Expressions, expression conversion

Power expressions (expressions with powers) and their transformation

In this article we will talk about converting expressions with powers. First, we will focus on transformations that are performed with expressions of any kind, including power expressions, such as opening parentheses and bringing similar terms. And then we will analyze the transformations inherent specifically in expressions with degrees: working with the base and exponent, using the properties of degrees, etc.

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What are power expressions?

The term “power expressions” practically does not appear in school mathematics textbooks, but it appears quite often in collections of problems, especially those intended for preparation for the Unified State Exam and the Unified State Exam, for example. After analyzing the tasks in which it is necessary to perform any actions with power expressions, it becomes clear that power expressions are understood as expressions containing powers in their entries. Therefore, you can accept the following definition for yourself:

Definition.

Power expressions are expressions containing degrees.

Let's give examples of power expressions. Moreover, we will present them according to how the development of views on from a degree with a natural exponent to a degree with a real exponent occurs.

As is known, first one gets acquainted with the power of a number with a natural exponent; at this stage, the first simplest power expressions of the type 3 2, 7 5 +1, (2+1) 5, (−0.1) 4, 3 a 2 appear −a+a 2 , x 3−1 , (a 2) 3 etc.

A little later, the power of a number with an integer exponent is studied, which leads to the appearance of power expressions with negative integer powers, like the following: 3 −2, , a −2 +2 b −3 +c 2 .

In high school they return to degrees. There a degree with a rational exponent is introduced, which entails the appearance of the corresponding power expressions: , , and so on. Finally, degrees with irrational exponents and expressions containing them are considered: , .

The matter is not limited to the listed power expressions: further the variable penetrates into the exponent, and, for example, the following expressions arise: 2 x 2 +1 or . And after getting acquainted with , expressions with powers and logarithms begin to appear, for example, x 2·lgx −5·x lgx.

So, we have dealt with the question of what power expressions represent. Next we will learn to convert them.

Main types of transformations of power expressions

With power expressions, you can perform any of the basic identity transformations of expressions. For example, you can open parentheses, replace numerical expressions with their values, add similar terms, etc. Naturally, in this case, it is necessary to follow the accepted procedure for performing actions. Let's give examples.

Example.

Calculate the value of the power expression 2 3 ·(4 2 −12) .

Solution.

According to the order of execution of actions, first perform the actions in brackets. There, firstly, we replace the power 4 2 with its value 16 (if necessary, see), and secondly, we calculate the difference 16−12=4. We have 2 3 ·(4 2 −12)=2 3 ·(16−12)=2 3 ·4.

In the resulting expression, we replace the power 2 3 with its value 8, after which we calculate the product 8·4=32. This is the desired value.

So, 2 3 ·(4 2 −12)=2 3 ·(16−12)=2 3 ·4=8·4=32.

Answer:

2 3 ·(4 2 −12)=32.

Example.

Simplify expressions with powers 3 a 4 b −7 −1+2 a 4 b −7.

Solution.

Obviously, this expression contains similar terms 3·a 4 ·b −7 and 2·a 4 ·b −7 , and we can present them: .

Answer:

3 a 4 b −7 −1+2 a 4 b −7 =5 a 4 b −7 −1.

Example.

Express an expression with powers as a product.

Solution.

You can cope with the task by representing the number 9 as a power of 3 2 and then using the formula for abbreviated multiplication - difference of squares:

Answer:

There are also a number of identical transformations inherent specifically in power expressions. We will analyze them further.

Working with base and exponent

There are degrees whose base and/or exponent are not just numbers or variables, but some expressions. As an example, we give the entries (2+0.3·7) 5−3.7 and (a·(a+1)−a 2) 2·(x+1) .

When working with such expressions, you can replace both the expression in the base of the degree and the expression in the exponent with an identically equal expression in the ODZ of its variables. In other words, according to the rules known to us, we can separately transform the base of the degree and separately the exponent. It is clear that as a result of this transformation, an expression will be obtained that is identically equal to the original one.

Such transformations allow us to simplify expressions with powers or achieve other goals we need. For example, in the power expression mentioned above (2+0.3 7) 5−3.7, you can perform operations with the numbers in the base and exponent, which will allow you to move to the power 4.1 1.3. And after opening the brackets and bringing similar terms to the base of the degree (a·(a+1)−a 2) 2·(x+1), we obtain a power expression of a simpler form a 2·(x+1) .

Using Degree Properties

One of the main tools for transforming expressions with powers is equalities that reflect . Let us recall the main ones. For any positive numbers a and b and arbitrary real numbers r and s, the following properties of powers are true:

• a r ·a s =a r+s ;
• a r:a s =a r−s ;
• (a·b) r =a r ·b r ;
• (a:b) r =a r:b r ;
• (a r) s =a r·s .

Note that for natural, integer, and positive exponents, the restrictions on the numbers a and b may not be so strict. For example, for natural numbers m and n the equality a m ·a n =a m+n is true not only for positive a, but also for negative a, and for a=0.

At school, the main focus when transforming power expressions is on the ability to choose the appropriate property and apply it correctly. In this case, the bases of degrees are usually positive, which allows the properties of degrees to be used without restrictions. The same applies to the transformation of expressions containing variables in the bases of powers - the range of permissible values ​​of variables is usually such that the bases take only positive values ​​on it, which allows you to freely use the properties of powers. In general, you need to constantly ask yourself whether it is possible to use any property of degrees in this case, because inaccurate use of properties can lead to a narrowing of the educational value and other troubles. These points are discussed in detail and with examples in the article transformation of expressions using properties of degrees. Here we will limit ourselves to considering a few simple examples.

Example.

Express the expression a 2.5 ·(a 2) −3:a −5.5 as a power with base a.

Solution.

First, we transform the second factor (a 2) −3 using the property of raising a power to a power: (a 2) −3 =a 2·(−3) =a −6. The original power expression will take the form a 2.5 ·a −6:a −5.5. Obviously, it remains to use the properties of multiplication and division of powers with the same base, we have
a 2.5 ·a −6:a −5.5 =
a 2.5−6:a −5.5 =a −3.5:a −5.5 =
a −3.5−(−5.5) =a 2 .

Answer:

a 2.5 ·(a 2) −3:a −5.5 =a 2.

Properties of powers when transforming power expressions are used both from left to right and from right to left.

Example.

Find the value of the power expression.

Solution.

The equality (a·b) r =a r ·b r, applied from right to left, allows us to move from the original expression to a product of the form and further. And when multiplying powers with the same bases, the exponents add up: .

It was possible to transform the original expression in another way:

Answer:

.

Example.

Given the power expression a 1.5 −a 0.5 −6, introduce a new variable t=a 0.5.

Solution.

The degree a 1.5 can be represented as a 0.5 3 and then, based on the property of the degree to the degree (a r) s =a r s, applied from right to left, transform it to the form (a 0.5) 3. Thus, a 1.5 −a 0.5 −6=(a 0.5) 3 −a 0.5 −6. Now it’s easy to introduce a new variable t=a 0.5, we get t 3 −t−6.

Answer:

t 3 −t−6 .

Converting fractions containing powers

Power expressions can contain or represent fractions with powers. Any of the basic transformations of fractions that are inherent in fractions of any kind are fully applicable to such fractions. That is, fractions that contain powers can be reduced, reduced to a new denominator, worked separately with their numerator and separately with the denominator, etc. To illustrate these words, consider solutions to several examples.

Example.

Simplify power expression .

Solution.

This power expression is a fraction. Let's work with its numerator and denominator. In the numerator we open the brackets and simplify the resulting expression using the properties of powers, and in the denominator we present similar terms:

And let’s also change the sign of the denominator by placing a minus in front of the fraction: .

Answer:

.

Reducing fractions containing powers to a new denominator is carried out similarly to reducing rational fractions to a new denominator. In this case, an additional factor is also found and the numerator and denominator of the fraction are multiplied by it. When performing this action, it is worth remembering that reduction to a new denominator can lead to a narrowing of the VA. To prevent this from happening, it is necessary that the additional factor does not go to zero for any values ​​of the variables from the ODZ variables for the original expression.

Example.

Reduce the fractions to a new denominator: a) to denominator a, b) to the denominator.

Solution.

a) In this case, it is quite easy to figure out which additional multiplier helps to achieve the desired result. This is a multiplier of a 0.3, since a 0.7 ·a 0.3 =a 0.7+0.3 =a. Note that in the range of permissible values ​​of the variable a (this is the set of all positive real numbers), the power of a 0.3 does not vanish, therefore, we have the right to multiply the numerator and denominator of a given fraction by this additional factor:

b) Taking a closer look at the denominator, you will find that

and multiplying this expression by will give the sum of cubes and , that is, . And this is the new denominator to which we need to reduce the original fraction.

This is how we found an additional factor. In the range of permissible values ​​of the variables x and y, the expression does not vanish, therefore, we can multiply the numerator and denominator of the fraction by it:

Answer:

A) , b) .

There is also nothing new in reducing fractions containing powers: the numerator and denominator are represented as a number of factors, and the same factors of the numerator and denominator are reduced.

Example.

Reduce the fraction: a) , b) .

Solution.

a) Firstly, the numerator and denominator can be reduced by the numbers 30 and 45, which is equal to 15. It is also obviously possible to perform a reduction by x 0.5 +1 and by . Here's what we have:

b) In this case, identical factors in the numerator and denominator are not immediately visible. To obtain them, you will have to perform preliminary transformations. In this case, they consist in factoring the denominator using the difference of squares formula:

Answer:

A)

b) .

Converting fractions to a new denominator and reducing fractions are mainly used to do things with fractions. Actions are performed according to known rules. When adding (subtracting) fractions, they are reduced to a common denominator, after which the numerators are added (subtracted), but the denominator remains the same. The result is a fraction whose numerator is the product of the numerators, and the denominator is the product of the denominators. Division by a fraction is multiplication by its inverse.

Example.

Follow the steps .

Solution.

First, we subtract the fractions in parentheses. To do this, we bring them to a common denominator, which is , after which we subtract the numerators:

Now we multiply the fractions:

Obviously, it is possible to reduce by a power of x 1/2, after which we have .

You can also simplify the power expression in the denominator by using the difference of squares formula: .

Answer:

Example.

Simplify the Power Expression .

Solution.

Obviously, this fraction can be reduced by (x 2.7 +1) 2, this gives the fraction . It is clear that something else needs to be done with the powers of X. To do this, we transform the resulting fraction into a product. This gives us the opportunity to take advantage of the property of dividing powers with the same bases: . And at the end of the process we move from the last product to the fraction.

Answer:

.

And let us also add that it is possible, and in many cases desirable, to transfer factors with negative exponents from the numerator to the denominator or from the denominator to the numerator, changing the sign of the exponent. Such transformations often simplify further actions. For example, a power expression can be replaced by .

Converting expressions with roots and powers

Often, in expressions in which some transformations are required, roots with fractional exponents are also present along with powers. To transform such an expression to the desired form, in most cases it is enough to go only to roots or only to powers. But since it is more convenient to work with powers, they usually move from roots to powers. However, it is advisable to carry out such a transition when the ODZ of variables for the original expression allows you to replace the roots with powers without the need to refer to the module or split the ODZ into several intervals (we discussed this in detail in the article transition from roots to powers and back After getting acquainted with the degree with a rational exponent a degree with an irrational exponent is introduced, which allows us to talk about a degree with an arbitrary real exponent. At this stage, it begins to be studied at school. exponential function, which is analytically given by a power, the base of which is a number, and the exponent is a variable. So we are faced with power expressions containing numbers in the base of the power, and in the exponent - expressions with variables, and naturally the need arises to perform transformations of such expressions.

It should be said that the transformation of expressions of the indicated type usually has to be performed when solving exponential equations And exponential inequalities, and these conversions are quite simple. In the overwhelming majority of cases, they are based on the properties of the degree and are aimed, for the most part, at introducing a new variable in the future. The equation will allow us to demonstrate them 5 2 x+1 −3 5 x 7 x −14 7 2 x−1 =0.

Firstly, powers, in the exponents of which is the sum of a certain variable (or expression with variables) and a number, are replaced by products. This applies to the first and last terms of the expression on the left side:
5 2 x 5 1 −3 5 x 7 x −14 7 2 x 7 −1 =0,
5 5 2 x −3 5 x 7 x −2 7 2 x =0.

Next, both sides of the equality are divided by the expression 7 2 x, which on the ODZ of the variable x for the original equation takes only positive values ​​(this is a standard technique for solving equations of this type, we are not talking about it now, so focus on subsequent transformations of expressions with powers ):

Now we can cancel fractions with powers, which gives .

Finally, the ratio of powers with the same exponents is replaced by powers of relations, resulting in the equation , which is equivalent . The transformations made allow us to introduce a new variable, which reduces the solution of the original exponential equation to the solution of a quadratic equation

I. V. Boykov, L. D. Romanova Collection of tasks for preparing for the Unified State Exam. Part 1. Penza 2003.

Why are degrees needed?

Where will you need them?

Why should you take the time to study them?

To find out EVERYTHING ABOUT DEGREES, read this article.

And, of course, knowledge of degrees will bring you closer to successfully passing the Unified State Exam.

And to admission to the university of your dreams!

Let's go... (Let's go!)

FIRST LEVEL

Exponentiation is a mathematical operation just like addition, subtraction, multiplication or division.

Now I will explain everything in human language using very simple examples. Be careful. The examples are elementary, but explain important things.

Let's start with addition.

There is nothing to explain here. You already know everything: there are eight of us. Each person has two bottles of cola. How much cola is there? That's right - 16 bottles.

Now multiplication.

The same example with cola can be written differently: . Mathematicians are cunning and lazy people. They first notice some patterns, and then figure out a way to “count” them faster. In our case, they noticed that each of the eight people had the same number of cola bottles and came up with a technique called multiplication. Agree, it is considered easier and faster than.

So, to count faster, easier and without errors, you just need to remember multiplication table. Of course, you can do everything slower, more difficult and with mistakes! But…

Here is the multiplication table. Repeat.

And another, more beautiful one:

What other clever counting tricks have lazy mathematicians come up with? Right - raising a number to a power.

Raising a number to a power

If you need to multiply a number by itself five times, then mathematicians say that you need to raise that number to the fifth power. For example, . Mathematicians remember that two to the fifth power is... And they solve such problems in their heads - faster, easier and without mistakes.

All you need to do is remember what is highlighted in color in the table of powers of numbers. Believe me, this will make your life a lot easier.

By the way, why is it called the second degree? square numbers, and the third - cube? What does it mean? Very good question. Now you will have both squares and cubes.

Real life example #1

Let's start with the square or the second power of the number.

Imagine a square pool measuring one meter by one meter. The pool is at your dacha. It's hot and I really want to swim. But... the pool has no bottom! You need to cover the bottom of the pool with tiles. How many tiles do you need? In order to determine this, you need to know the bottom area of ​​the pool.

You can simply calculate by pointing your finger that the bottom of the pool consists of meter by meter cubes. If you have tiles one meter by one meter, you will need pieces. It's easy... But where have you seen such tiles? The tile will most likely be cm by cm. And then you will be tortured by “counting with your finger.” Then you have to multiply. So, on one side of the bottom of the pool we will fit tiles (pieces) and on the other, too, tiles. Multiply by and you get tiles ().

Did you notice that to determine the area of ​​the pool bottom we multiplied the same number by itself? What does it mean? Since we are multiplying the same number, we can use the “exponentiation” technique. (Of course, when you have only two numbers, you still need to multiply them or raise them to a power. But if you have a lot of them, then raising them to a power is much easier and there are also fewer errors in calculations. For the Unified State Exam, this is very important).
So, thirty to the second power will be (). Or we can say that thirty squared will be. In other words, the second power of a number can always be represented as a square. And vice versa, if you see a square, it is ALWAYS the second power of some number. A square is an image of the second power of a number.

Real life example #2

Here's a task for you: count how many squares there are on the chessboard using the square of the number... On one side of the cells and on the other too. To calculate their number, you need to multiply eight by eight or... if you notice that a chessboard is a square with a side, then you can square eight. You will get cells. () So?

Real life example #3

Now the cube or the third power of a number. The same pool. But now you need to find out how much water will have to be poured into this pool. You need to calculate the volume. (Volumes and liquids, by the way, are measured in cubic meters. Unexpected, right?) Draw a pool: the bottom is a meter in size and a meter deep, and try to calculate how many cubes measuring a meter by a meter will fit into your pool.

Just point your finger and count! One, two, three, four...twenty-two, twenty-three...How many did you get? Not lost? Is it difficult to count with your finger? So that! Take an example from mathematicians. They are lazy, so they noticed that in order to calculate the volume of the pool, you need to multiply its length, width and height by each other. In our case, the volume of the pool will be equal to cubes... Easier, right?

Now imagine how lazy and cunning mathematicians are if they simplified this too. We reduced everything to one action. They noticed that the length, width and height are equal and that the same number is multiplied by itself... What does this mean? This means you can take advantage of the degree. So, what you once counted with your finger, they do in one action: three cubed is equal. It is written like this: .

All that remains is remember the table of degrees. Unless, of course, you are as lazy and cunning as mathematicians. If you like to work hard and make mistakes, you can continue to count with your finger.

Well, to finally convince you that degrees were invented by quitters and cunning people to solve their life problems, and not to create problems for you, here are a couple more examples from life.

Real life example #4

You have a million rubles. At the beginning of each year, for every million you make, you make another million. That is, every million of yours doubles at the beginning of each year. How much money will you have in years? If you are sitting now and “counting with your finger,” then you are a very hardworking person and... stupid. But most likely you will give an answer in a couple of seconds, because you are smart! So, in the first year - two multiplied by two... in the second year - what happened, by two more, in the third year... Stop! You noticed that the number is multiplied by itself times. So two to the fifth power is a million! Now imagine that you have a competition and the one who can count the fastest will get these millions... It’s worth remembering the powers of numbers, don’t you think?

Real life example #5

You have a million. At the beginning of each year, you earn two more for every million. Great isn't it? Every million is tripled. How much money will you have in a year? Let's count. The first year - multiply by, then the result by another... It’s already boring, because you already understood everything: three is multiplied by itself times. So to the fourth power it is equal to a million. You just have to remember that three to the fourth power is or.

Now you know that by raising a number to a power you will make your life a lot easier. Let's take a further look at what you can do with degrees and what you need to know about them.

Terms and concepts... so as not to get confused

So, first, let's define the concepts. What do you think, what is an exponent? It's very simple - it's the number that is "at the top" of the power of the number. Not scientific, but clear and easy to remember...

Well, at the same time, what such a degree basis? Even simpler - this is the number that is located below, at the base.

Here's a drawing for good measure.

Well, in general terms, in order to generalize and remember better... A degree with a base “ ” and an exponent “ ” is read as “to the degree” and is written as follows:

Power of a number with natural exponent

You probably already guessed: because the exponent is a natural number. Yes, but what is it natural number? Elementary! Natural numbers are those numbers that are used in counting when listing objects: one, two, three... When we count objects, we do not say: “minus five,” “minus six,” “minus seven.” We also do not say: “one third”, or “zero point five”. These are not natural numbers. What numbers do you think these are?

Numbers like “minus five”, “minus six”, “minus seven” refer to whole numbers. In general, integers include all natural numbers, numbers opposite to natural numbers (that is, taken with a minus sign), and number. Zero is easy to understand - it is when there is nothing. What do negative (“minus”) numbers mean? But they were invented primarily to indicate debts: if you have a balance on your phone in rubles, this means that you owe the operator rubles.

All fractions are rational numbers. How did they arise, do you think? Very simple. Several thousand years ago, our ancestors discovered that they lacked natural numbers to measure length, weight, area, etc. And they came up with rational numbers... Interesting, isn't it?

There are also irrational numbers. What are these numbers? In short, it's an infinite decimal fraction. For example, if you divide the circumference of a circle by its diameter, you get an irrational number.

Summary:

Let us define the concept of a degree whose exponent is a natural number (i.e., integer and positive).

1. Any number to the first power is equal to itself:
2. To square a number means to multiply it by itself:
3. To cube a number means to multiply it by itself three times:

Definition. Raising a number to a natural power means multiplying the number by itself times:
.

Properties of degrees

Where did these properties come from? I will show you now.

Let's see: what is it And ?

A-priory:

How many multipliers are there in total?

It’s very simple: we added multipliers to the factors, and the result is multipliers.

But by definition, this is a power of a number with an exponent, that is: , which is what needed to be proven.

Example: Simplify the expression.

Solution:

Example: Simplify the expression.

Solution: It is important to note that in our rule Necessarily there must be the same reasons!
Therefore, we combine the powers with the base, but it remains a separate factor:

only for the product of powers!

Under no circumstances can you write that.

2. that's it th power of a number

Just as with the previous property, let us turn to the definition of degree:

It turns out that the expression is multiplied by itself times, that is, according to the definition, this is the th power of the number:

In essence, this can be called “taking the indicator out of brackets.” But you can never do this in total:

Let's remember the abbreviated multiplication formulas: how many times did we want to write?

But this is not true, after all.

Power with negative base

Up to this point, we have only discussed what the exponent should be.

But what should be the basis?

In powers of natural indicator the basis may be any number. Indeed, we can multiply any numbers by each other, be they positive, negative, or even.

Let's think about which signs ("" or "") will have degrees of positive and negative numbers?

For example, is the number positive or negative? A? ? With the first one, everything is clear: no matter how many positive numbers we multiply by each other, the result will be positive.

But the negative ones are a little more interesting. We remember the simple rule from 6th grade: “minus for minus gives a plus.” That is, or. But if we multiply by, it works.

Determine for yourself what sign the following expressions will have:

1)	2)	3)
4)	5)	6)

Did you manage?

Here are the answers: In the first four examples, I hope everything is clear? We simply look at the base and exponent and apply the appropriate rule.

In example 5) everything is also not as scary as it seems: after all, it doesn’t matter what the base is equal to - the degree is even, which means the result will always be positive.

Well, except when the base is zero. The base is not equal, is it? Obviously not, since (because).

Example 6) is no longer so simple!

6 examples to practice

Analysis of the solution 6 examples

Whole we call the natural numbers, their opposites (that is, taken with the " " sign) and the number.

positive integer, and it is no different from natural, then everything looks exactly like in the previous section.

Now let's look at new cases. Let's start with an indicator equal to.

Any number to the zero power is equal to one:

As always, let us ask ourselves: why is this so?

Let's consider some degree with a base. Take, for example, and multiply by:

So, we multiplied the number by, and we got the same thing as it was - . What number should you multiply by so that nothing changes? That's right, on. Means.

We can do the same with an arbitrary number:

Let's repeat the rule:

Any number to the zero power is equal to one.

But there are exceptions to many rules. And here it is also there - this is a number (as a base).

On the one hand, it must be equal to any degree - no matter how much you multiply zero by itself, you will still get zero, this is clear. But on the other hand, like any number to the zero power, it must be equal. So how much of this is true? The mathematicians decided not to get involved and refused to raise zero to the zero power. That is, now we cannot not only divide by zero, but also raise it to the zero power.

Let's move on. In addition to natural numbers and numbers, integers also include negative numbers. To understand what a negative power is, let’s do as last time: multiply some normal number by the same number to a negative power:

From here it is easy to express what you are looking for:

Now let’s extend the resulting rule to an arbitrary degree:

So, let's formulate a rule:

A number with a negative power is the reciprocal of the same number with a positive power. But at the same time The base cannot be null:(because you can’t divide by).

Let's summarize:

Tasks for independent solution:

Well, as usual, examples for independent solutions:

Analysis of problems for independent solution:

I know, I know, the numbers are scary, but on the Unified State Exam you have to be prepared for anything! Solve these examples or analyze their solutions if you couldn’t solve them and you will learn to cope with them easily in the exam!

Let's continue to expand the range of numbers “suitable” as an exponent.

Now let's consider rational numbers. What numbers are called rational?

Answer: everything that can be represented as a fraction, where and are integers, and.

To understand what it is "fractional degree", consider the fraction:

Let's raise both sides of the equation to a power:

Now let's remember the rule about "degree to degree":

What number must be raised to a power to get?

This formulation is the definition of the root of the th degree.

Let me remind you: the root of the th power of a number () is a number that, when raised to a power, is equal to.

That is, the root of the th power is the inverse operation of raising to a power: .

It turns out that. Obviously, this special case can be expanded: .

Now we add the numerator: what is it? The answer is easy to obtain using the power-to-power rule:

But can the base be any number? After all, the root cannot be extracted from all numbers.

None!

Let us remember the rule: any number raised to an even power is a positive number. That is, it is impossible to extract even roots from negative numbers!

This means that such numbers cannot be raised to a fractional power with an even denominator, that is, the expression does not make sense.

What about the expression?

But here a problem arises.

The number can be represented in the form of other, reducible fractions, for example, or.

And it turns out that it exists, but does not exist, but these are just two different records of the same number.

Or another example: once, then you can write it down. But if we write down the indicator differently, we will get into trouble again: (that is, we got a completely different result!).

To avoid such paradoxes, we consider only positive base exponent with fractional exponent.

So if:

• - natural number;
• - integer;

Examples:

Rational exponents are very useful for transforming expressions with roots, for example:

5 examples to practice

Analysis of 5 examples for training

Well, now comes the hardest part. Now we'll figure it out degree with irrational exponent.

All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, with the exception

After all, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are integers (that is, irrational numbers are all real numbers except rational ones).

When studying degrees with natural, integer and rational exponents, each time we created a certain “image”, “analogy”, or description in more familiar terms.

For example, a degree with a natural exponent is a number multiplied by itself several times;

...number to the zeroth power- this is, as it were, a number multiplied by itself once, that is, they have not yet begun to multiply it, which means that the number itself has not even appeared yet - therefore the result is only a certain “blank number”, namely a number;

...negative integer degree- it’s as if some “reverse process” had occurred, that is, the number was not multiplied by itself, but divided.

By the way, in science a degree with a complex exponent is often used, that is, the exponent is not even a real number.

But at school we don’t think about such difficulties; you will have the opportunity to comprehend these new concepts at the institute.

WHERE WE ARE SURE YOU WILL GO! (if you learn to solve such examples :))

For example:

Decide for yourself:

Analysis of solutions:

1. Let's start with the rule for raising a power to a power, which is already usual for us:

ADVANCED LEVEL

Determination of degree

A degree is an expression of the form: , where:

• — degree base;
• - exponent.

Degree with natural indicator (n = 1, 2, 3,...)

Raising a number to the natural power n means multiplying the number by itself times:

Degree with an integer exponent (0, ±1, ±2,...)

If the exponent is positive integer number:

Construction to the zero degree:

The expression is indefinite, because, on the one hand, to any degree is this, and on the other hand, any number to the th degree is this.

If the exponent is negative integer number:

(because you can’t divide by).

Once again about zeros: the expression is not defined in the case. If, then.

Examples:

Power with rational exponent

• - natural number;
• - integer;

Examples:

Properties of degrees

To make it easier to solve problems, let’s try to understand: where did these properties come from? Let's prove them.

Let's see: what is and?

A-priory:

So, on the right side of this expression we get the following product:

But by definition it is a power of a number with an exponent, that is:

Q.E.D.

Example : Simplify the expression.

Solution : .

Example : Simplify the expression.

Solution : It is important to note that in our rule Necessarily there must be the same reasons. Therefore, we combine the powers with the base, but it remains a separate factor:

Another important note: this rule - only for product of powers!

Under no circumstances can you write that.

Just as with the previous property, let us turn to the definition of degree:

Let's regroup this work like this:

It turns out that the expression is multiplied by itself times, that is, according to the definition, this is the th power of the number:

In essence, this can be called “taking the indicator out of brackets.” But you can never do this in total: !

Let's remember the abbreviated multiplication formulas: how many times did we want to write? But this is not true, after all.

Power with a negative base.

Up to this point we have only discussed what it should be like index degrees. But what should be the basis? In powers of natural indicator the basis may be any number .

Indeed, we can multiply any numbers by each other, be they positive, negative, or even. Let's think about which signs ("" or "") will have degrees of positive and negative numbers?

For example, is the number positive or negative? A? ?

With the first one, everything is clear: no matter how many positive numbers we multiply by each other, the result will be positive.

But the negative ones are a little more interesting. We remember the simple rule from 6th grade: “minus for minus gives a plus.” That is, or. But if we multiply by (), we get - .

And so on ad infinitum: with each subsequent multiplication the sign will change. The following simple rules can be formulated:

1. even degree, - number positive.
2. Negative number raised to odd degree, - number negative.
3. A positive number to any degree is a positive number.
4. Zero to any power is equal to zero.

Determine for yourself what sign the following expressions will have:

1.	2.	3.
4.	5.	6.

Did you manage? Here are the answers:

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In the first four examples, I hope everything is clear? We simply look at the base and exponent and apply the appropriate rule.

In example 5) everything is also not as scary as it seems: after all, it doesn’t matter what the base is equal to - the degree is even, which means the result will always be positive. Well, except when the base is zero. The base is not equal, is it? Obviously not, since (because).

Example 6) is no longer so simple. Here you need to find out which is less: or? If we remember that, it becomes clear that, which means the base is less than zero. That is, we apply rule 2: the result will be negative.

And again we use the definition of degree:

Everything is as usual - we write down the definition of degrees and divide them by each other, divide them into pairs and get:

Before we look at the last rule, let's solve a few examples.

Calculate the expressions:

Solutions :

Let's go back to the example:

And again the formula:

So now the last rule:

How will we prove it? Of course, as usual: let’s expand on the concept of degree and simplify it:

Well, now let's open the brackets. How many letters are there in total? times by multipliers - what does this remind you of? This is nothing more than a definition of an operation multiplication: There were only multipliers there. That is, this, by definition, is a power of a number with an exponent:

Example:

Degree with irrational exponent

In addition to information about degrees for the average level, we will analyze the degree with an irrational exponent. All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, with the exception - after all, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are integers (that is, irrational numbers are all real numbers except rational numbers).

When studying degrees with natural, integer and rational exponents, each time we created a certain “image”, “analogy”, or description in more familiar terms. For example, a degree with a natural exponent is a number multiplied by itself several times; a number to the zero power is, as it were, a number multiplied by itself once, that is, they have not yet begun to multiply it, which means that the number itself has not even appeared yet - therefore the result is only a certain “blank number”, namely a number; a degree with an integer negative exponent - it’s as if some “reverse process” had occurred, that is, the number was not multiplied by itself, but divided.

It is extremely difficult to imagine a degree with an irrational exponent (just as it is difficult to imagine a 4-dimensional space). It is rather a purely mathematical object that mathematicians created to extend the concept of degree to the entire space of numbers.

By the way, in science a degree with a complex exponent is often used, that is, the exponent is not even a real number. But at school we don’t think about such difficulties; you will have the opportunity to comprehend these new concepts at the institute.

So what do we do if we see an irrational exponent? We are trying our best to get rid of it! :)

For example:

Decide for yourself:

1)

2)

3)

Answers:

SUMMARY OF THE SECTION AND BASIC FORMULAS

Degree called an expression of the form: , where:

Degree with an integer exponent

a degree whose exponent is a natural number (i.e., integer and positive).

Power with rational exponent

degree, the exponent of which is negative and fractional numbers.

Degree with irrational exponent

a degree whose exponent is an infinite decimal fraction or root.

Properties of degrees

Features of degrees.

• Negative number raised to even degree, - number positive.
• Negative number raised to odd degree, - number negative.
• A positive number to any degree is a positive number.
• Zero is equal to any power.
• Any number to the zero power is equal.

NOW YOU HAVE THE WORD...

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In this material we will look at what a power of a number is. In addition to the basic definitions, we will formulate what powers with natural, integer, rational and irrational exponents are. As always, all concepts will be illustrated with example problems.

Yandex.RTB R-A-339285-1

First, let's formulate the basic definition of a degree with a natural exponent. To do this, we need to remember the basic rules of multiplication. Let us clarify in advance that for now we will take a real number as a base (denoted by the letter a), and a natural number as an indicator (denoted by the letter n).

Definition 1

The power of a number a with natural exponent n is the product of the nth number of factors, each of which is equal to the number a. The degree is written like this: a n, and in the form of a formula its composition can be represented as follows:

For example, if the exponent is 1 and the base is a, then the first power of a is written as a 1. Given that a is the value of the factor and 1 is the number of factors, we can conclude that a 1 = a.

In general, we can say that a degree is a convenient form of writing a large number of equal factors. So, a record of the form 8 8 8 8 can be shortened to 8 4 . In much the same way, the product helps us avoid writing a large number of terms (8 + 8 + 8 + 8 = 8 · 4); We have already discussed this in the article devoted to the multiplication of natural numbers.

How to correctly read the degree entry? The generally accepted option is “a to the power of n”. Or you can say “nth power of a” or “anth power”. If, say, in the example we encountered the entry 8 12 , we can read "8 to the 12th power", "8 to the power of 12" or "12th power of 8".

The second and third powers of numbers have their own established names: square and cube. If we see the second power, for example, the number 7 (7 2), then we can say “7 squared” or “square of the number 7”. Similarly, the third degree is read like this: 5 3 - this is the “cube of the number 5” or “5 cubed.” However, you can also use the standard formulation “to the second/third power”; this will not be a mistake.

Example 1

Let's look at an example of a degree with a natural exponent: for 5 7 five will be the base, and seven will be the exponent.

The base does not have to be an integer: for the degree (4 , 32) 9 the base will be the fraction 4, 32, and the exponent will be nine. Pay attention to the parentheses: this notation is made for all powers whose bases differ from natural numbers.

For example: 1 2 3, (- 3) 12, - 2 3 5 2, 2, 4 35 5, 7 3.

What are parentheses for? They help avoid errors in calculations. Let's say we have two entries: (− 2) 3 And − 2 3 . The first of these means a negative number minus two raised to a power with a natural exponent of three; the second is the number corresponding to the opposite value of the degree 2 3 .

Sometimes in books you can find a slightly different spelling of the power of a number - a^n(where a is the base and n is the exponent). That is, 4^9 is the same as 4 9 . If n is a multi-digit number, it is placed in parentheses. For example, 15 ^ (21) , (− 3 , 1) ^ (156) . But we will use the notation a n as more common.

It’s easy to guess how to calculate the value of an exponent with a natural exponent from its definition: you just need to multiply a nth number of times. We wrote more about this in another article.

The concept of degree is the inverse of another mathematical concept - the root of a number. If we know the value of the power and the exponent, we can calculate its base. The degree has some specific properties that are useful for solving problems, which we discussed in a separate material.

Exponents can include not only natural numbers, but also any integer values ​​in general, including negative ones and zeros, because they also belong to the set of integers.

Definition 2

The power of a number with a positive integer exponent can be represented as a formula: .

In this case, n is any positive integer.

Let's understand the concept of zero degree. To do this, we use an approach that takes into account the quotient property for powers with equal bases. It is formulated like this:

Definition 3

Equality a m: a n = a m − n will be true under the following conditions: m and n are natural numbers, m< n , a ≠ 0 .

The last condition is important because it avoids division by zero. If the values ​​of m and n are equal, then we get the following result: a n: a n = a n − n = a 0

But at the same time a n: a n = 1 is the quotient of equal numbers a n and a. It turns out that the zero power of any non-zero number is equal to one.

However, such a proof does not apply to zero to the zeroth power. To do this, we need another property of powers - the property of products of powers with equal bases. It looks like this: a m · a n = a m + n .

If n is equal to 0, then a m · a 0 = a m(this equality also proves to us that a 0 = 1). But if and is also equal to zero, our equality takes the form 0 m · 0 0 = 0 m, It will be true for any natural value of n, and it does not matter what exactly the value of the degree is 0 0 , that is, it can be equal to any number, and this will not affect the accuracy of the equality. Therefore, a notation of the form 0 0 does not have its own special meaning, and we will not attribute it to it.

If desired, it is easy to check that a 0 = 1 converges with the degree property (a m) n = a m n provided that the base of the degree is not zero. Thus, the power of any non-zero number with exponent zero is one.

Example 2

Let's look at an example with specific numbers: So, 5 0 - unit, (33 , 3) 0 = 1 , - 4 5 9 0 = 1 , and the value 0 0 undefined.

After the zero degree, we just have to figure out what a negative degree is. To do this, we need the same property of the product of powers with equal bases that we already used above: a m · a n = a m + n.

Let us introduce the condition: m = − n, then a should not be equal to zero. It follows that a − n · a n = a − n + n = a 0 = 1. It turns out that a n and a−n we have mutually reciprocal numbers.

As a result, a to the negative whole power is nothing more than the fraction 1 a n.

This formulation confirms that for a degree with an integer negative exponent, all the same properties are valid that a degree with a natural exponent has (provided that the base is not equal to zero).

Example 3

A power a with a negative integer exponent n can be represented as a fraction 1 a n . Thus, a - n = 1 a n subject to a ≠ 0 and n is any natural number.

Let us illustrate our idea with specific examples:

Example 4

3 - 2 = 1 3 2 , (- 4 . 2) - 5 = 1 (- 4 . 2) 5 , 11 37 - 1 = 1 11 37 1

In the last part of the paragraph, we will try to depict everything that has been said clearly in one formula:

Definition 4

The power of a number with a natural exponent z is: a z = a z, e with l and z - positive integer 1, z = 0 and a ≠ 0, (for z = 0 and a = 0 the result is 0 0, the values ​​of the expression 0 0 are not is defined) 1 a z, if and z is a negative integer and a ≠ 0 ( if z is a negative integer and a = 0 you get 0 z, egoz the value is undetermined)

What are powers with a rational exponent?

We examined cases when the exponent contains an integer. However, you can raise a number to a power even when its exponent contains a fractional number. This is called a power with a rational exponent. In this section we will prove that it has the same properties as other powers.

What are rational numbers? Their set includes both whole and fractional numbers, and fractional numbers can be represented as ordinary fractions (both positive and negative). Let us formulate the definition of the power of a number a with a fractional exponent m / n, where n is a natural number and m is an integer.

We have some degree with a fractional exponent a m n . In order for the power to power property to hold, the equality a m n n = a m n · n = a m must be true.

Given the definition of the nth root and that a m n n = a m, we can accept the condition a m n = a m n if a m n makes sense for the given values ​​of m, n and a.

The above properties of a degree with an integer exponent will be true under the condition a m n = a m n .

The main conclusion from our reasoning is this: the power of a certain number a with a fractional exponent m / n is the nth root of the number a to the power m. This is true if, for given values ​​of m, n and a, the expression a m n remains meaningful.

1. We can limit the value of the base of the degree: let's take a, which for positive values ​​of m will be greater than or equal to 0, and for negative values ​​- strictly less (since for m ≤ 0 we get 0 m, but such a degree is not defined). In this case, the definition of a degree with a fractional exponent will look like this:

A power with a fractional exponent m/n for some positive number a is the nth root of a raised to the power m. This can be expressed as a formula:

For a power with a zero base, this provision is also suitable, but only if its exponent is a positive number.

A power with a base zero and a fractional positive exponent m/n can be expressed as

0 m n = 0 m n = 0 provided m is a positive integer and n is a natural number.

For a negative ratio m n< 0 степень не определяется, т.е. такая запись смысла не имеет.

Let's note one point. Since we introduced the condition that a is greater than or equal to zero, we ended up discarding some cases.

The expression a m n sometimes still makes sense for some negative values ​​of a and some m. Thus, the correct entries are (- 5) 2 3, (- 1, 2) 5 7, - 1 2 - 8 4, in which the base is negative.

2. The second approach is to consider separately the root a m n with even and odd exponents. Then we will need to introduce one more condition: the degree a, in the exponent of which there is a reducible ordinary fraction, is considered to be the degree a, in the exponent of which there is the corresponding irreducible fraction. Later we will explain why we need this condition and why it is so important. Thus, if we have the notation a m · k n · k , then we can reduce it to a m n and simplify the calculations.

If n is an odd number and the value of m is positive and a is any non-negative number, then a m n makes sense. The condition for a to be non-negative is necessary because a root of an even degree cannot be extracted from a negative number. If the value of m is positive, then a can be both negative and zero, because The odd root can be taken from any real number.

Let's combine all the above definitions in one entry:

Here m/n means an irreducible fraction, m is any integer, and n is any natural number.

Definition 5

For any ordinary reducible fraction m · k n · k the degree can be replaced by a m n .

The power of a number a with an irreducible fractional exponent m / n – can be expressed as a m n in the following cases: - for any real a, positive integer values ​​m and odd natural values ​​n. Example: 2 5 3 = 2 5 3, (- 5, 1) 2 7 = (- 5, 1) - 2 7, 0 5 19 = 0 5 19.

For any non-zero real a, negative integer values ​​of m and odd values ​​of n, for example, 2 - 5 3 = 2 - 5 3, (- 5, 1) - 2 7 = (- 5, 1) - 2 7

For any non-negative a, positive integer m and even n, for example, 2 1 4 = 2 1 4, (5, 1) 3 2 = (5, 1) 3, 0 7 18 = 0 7 18.

For any positive a, negative integer m and even n, for example, 2 - 1 4 = 2 - 1 4, (5, 1) - 3 2 = (5, 1) - 3, .

In the case of other values, the degree with a fractional exponent is not determined. Examples of such degrees: - 2 11 6, - 2 1 2 3 2, 0 - 2 5.

Now let’s explain the importance of the condition discussed above: why replace a fraction with a reducible exponent with a fraction with an irreducible exponent. If we had not done this, we would have had the following situations, say, 6/10 = 3/5. Then it should be true (- 1) 6 10 = - 1 3 5 , but - 1 6 10 = (- 1) 6 10 = 1 10 = 1 10 10 = 1 , and (- 1) 3 5 = (- 1) 3 5 = - 1 5 = - 1 5 5 = - 1 .

The definition of a degree with a fractional exponent, which we presented first, is more convenient to use in practice than the second, so we will continue to use it.

Definition 6

Thus, the power of a positive number a with a fractional exponent m/n is defined as 0 m n = 0 m n = 0. In case of negative a the entry a m n does not make sense. Power of zero for positive fractional exponents m/n is defined as 0 m n = 0 m n = 0 , for negative fractional exponents we do not define the degree of zero.

In conclusions, we note that you can write any fractional indicator both as a mixed number and as a decimal fraction: 5 1, 7, 3 2 5 - 2 3 7.

When calculating, it is better to replace the exponent with an ordinary fraction and then use the definition of exponent with a fractional exponent. For the examples above we get:

5 1 , 7 = 5 17 10 = 5 7 10 3 2 5 - 2 3 7 = 3 2 5 - 17 7 = 3 2 5 - 17 7

What are powers with irrational and real exponents?

What are real numbers? Their set includes both rational and irrational numbers. Therefore, in order to understand what a degree with a real exponent is, we need to define degrees with rational and irrational exponents. We have already mentioned rational ones above. Let's deal with irrational indicators step by step.

Example 5

Let's assume that we have an irrational number a and a sequence of its decimal approximations a 0 , a 1 , a 2 , . . . . For example, let's take the value a = 1.67175331. . . , Then

a 0 = 1, 6, a 1 = 1, 67, a 2 = 1, 671, . . . , a 0 = 1.67, a 1 = 1.6717, a 2 = 1.671753, . . .

We can associate the sequence of approximations with the sequence of degrees a a 0 , a a 1 , a a 2 , . . . . If we remember what we said earlier about raising numbers to rational powers, then we can calculate the values ​​of these powers ourselves.

Let's take for example a = 3, then a a 0 = 3 1, 67, a a 1 = 3 1, 6717, a a 2 = 3 1, 671753, . . . etc.

The sequence of powers can be reduced to a number, which will be the value of the power with base a and irrational exponent a. As a result: a degree with an irrational exponent of the form 3 1, 67175331. . can be reduced to the number 6, 27.

Definition 7

The power of a positive number a with an irrational exponent a is written as a a . Its value is the limit of the sequence a a 0 , a a 1 , a a 2 , . . . , where a 0 , a 1 , a 2 , . . . are successive decimal approximations of the irrational number a. A degree with a zero base can also be defined for positive irrational exponents, with 0 a = 0 So, 0 6 = 0, 0 21 3 3 = 0. But this cannot be done for negative ones, since, for example, the value 0 - 5, 0 - 2 π is not defined. A unit raised to any irrational power remains a unit, for example, and 1 2, 1 5 in 2 and 1 - 5 will be equal to 1.

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Continuing the conversation about the power of a number, it is logical to figure out how to find the value of the power. This process is called exponentiation. In this article we will study how exponentiation is performed, while we will touch on all possible exponents - natural, integer, rational and irrational. And according to tradition, we will consider in detail solutions to examples of raising numbers to various powers.

Page navigation.

What does "exponentiation" mean?

Let's start by explaining what is called exponentiation. Here is the relevant definition.

Definition.

Exponentiation- this is finding the value of the power of a number.

Thus, finding the value of the power of a number a with exponent r and raising the number a to the power r are the same thing. For example, if the task is “calculate the value of the power (0.5) 5,” then it can be reformulated as follows: “Raise the number 0.5 to the power 5.”

Now you can go directly to the rules by which exponentiation is performed.

Raising a number to a natural power

In practice, equality based on is usually applied in the form . That is, when raising a number a to a fractional power m/n, the nth root of the number a is first taken, after which the resulting result is raised to an integer power m.

Let's look at solutions to examples of raising to a fractional power.

Example.

Calculate the value of the degree.

Solution.

We will show two solutions.

First way. By definition of a degree with a fractional exponent. We calculate the value of the degree under the root sign, and then extract the cube root: .

Second way. By the definition of a degree with a fractional exponent and based on the properties of the roots, the following equalities are true: . Now we extract the root , finally, we raise it to an integer power .

Obviously, the obtained results of raising to a fractional power coincide.

Answer:

Note that a fractional exponent can be written as a decimal fraction or a mixed number, in these cases it should be replaced with the corresponding ordinary fraction, and then raised to a power.

Example.

Calculate (44.89) 2.5.

Solution.

Let's write the exponent in the form of an ordinary fraction (if necessary, see the article): . Now we perform the raising to a fractional power:

Answer:

(44,89) 2,5 =13 501,25107 .

It should also be said that raising numbers to rational powers is a rather labor-intensive process (especially when the numerator and denominator of the fractional exponent contain sufficiently large numbers), which is usually carried out using computer technology.

To conclude this point, let us dwell on raising the number zero to a fractional power. We gave the following meaning to the fractional power of zero of the form: when we have , and at zero to the m/n power is not defined. So, zero to a fractional positive power is zero, for example, . And zero in a fractional negative power does not make sense, for example, the expressions 0 -4.3 do not make sense.

Raising to an irrational power

Sometimes it becomes necessary to find out the value of the power of a number with an irrational exponent. In this case, for practical purposes it is usually sufficient to obtain the value of the degree accurate to a certain sign. Let us immediately note that in practice this value is calculated using electronic computers, since raising it to an irrational power manually requires a large number of cumbersome calculations. But we will still describe in general terms the essence of the actions.

To obtain an approximate value of the power of a number a with an irrational exponent, some decimal approximation of the exponent is taken and the value of the power is calculated. This value is an approximate value of the power of the number a with an irrational exponent. The more accurate the decimal approximation of a number is taken initially, the more accurate the value of the degree will be obtained in the end.

As an example, let's calculate the approximate value of the power of 2 1.174367... . Let's take the following decimal approximation of the irrational exponent: . Now we raise 2 to the rational power 1.17 (we described the essence of this process in the previous paragraph), we get 2 1.17 ≈2.250116. Thus, 2 1,174367... ≈2 1,17 ≈2,250116 . If we take a more accurate decimal approximation of the irrational exponent, for example, then we obtain a more accurate value of the original exponent: 2 1,174367... ≈2 1,1743 ≈2,256833 .

Bibliography.

• Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics textbook for 5th grade. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 7th grade. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8th grade. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 9th grade. educational institutions.
• Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the beginnings of analysis: Textbook for grades 10 - 11 of general education institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for those entering technical schools).

The power is used to simplify the operation of multiplying a number by itself. For example, instead of writing, you can write 4 5 (\displaystyle 4^(5))(an explanation for this transition is given in the first section of this article). Degrees make it easier to write long or complex expressions or equations; powers are also easy to add and subtract, resulting in a simplified expression or equation (for example, 4 2 ∗ 4 3 = 4 5 (\displaystyle 4^(2)*4^(3)=4^(5))).

Note: if you need to solve an exponential equation (in such an equation the unknown is in the exponent), read.

Steps

Solving simple problems with degrees

Multiply the base of the exponent by itself a number of times equal to the exponent. If you need to solve a power problem by hand, rewrite the power as a multiplication operation, where the base of the power is multiplied by itself. For example, given a degree 3 4 (\displaystyle 3^(4)). In this case, the base of power 3 must be multiplied by itself 4 times: 3 ∗ 3 ∗ 3 ∗ 3 (\displaystyle 3*3*3*3). Here are other examples:

First, multiply the first two numbers. For example, 4 5 (\displaystyle 4^(5)) = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4*4*4*4*4). Don't worry - the calculation process is not as complicated as it seems at first glance. First multiply the first two fours and then replace them with the result. Like this:

Multiply the result (16 in our example) by the next number. Each subsequent result will increase proportionally. In our example, multiply 16 by 4. Like this:

Solve the following problems. Check your answer using a calculator.

On your calculator, look for the key labeled "exp" or " x n (\displaystyle x^(n)) ", or "^". Using this key you will raise a number to a power. It is almost impossible to calculate a degree with a large indicator manually (for example, the degree 9 15 (\displaystyle 9^(15))), but the calculator can easily cope with this task. In Windows 7, the standard calculator can be switched to engineering mode; To do this, click “View” -> “Engineering”. To switch to normal mode, click “View” -> “Normal”.

• Check the received answer using a search engine (Google or Yandex). Using the "^" key on your computer keyboard, enter the expression into the search engine, which will instantly display the correct answer (and perhaps suggest similar expressions to study).

Addition, subtraction, multiplication of powers

1. You can add and subtract degrees only if they have the same bases. If you need to add powers with the same bases and exponents, then you can replace the addition operation with the multiplication operation. For example, given the expression 4 5 + 4 5 (\displaystyle 4^(5)+4^(5)). Remember that the degree 4 5 (\displaystyle 4^(5)) can be represented in the form 1 ∗ 4 5 (\displaystyle 1*4^(5)); Thus, 4 5 + 4 5 = 1 ∗ 4 5 + 1 ∗ 4 5 = 2 ∗ 4 5 (\displaystyle 4^(5)+4^(5)=1*4^(5)+1*4^(5) =2*4^(5))(where 1 +1 =2). That is, count the number of similar degrees, and then multiply that degree and this number. In our example, raise 4 to the fifth power, and then multiply the resulting result by 2. Remember that the addition operation can be replaced by the multiplication operation, for example, 3 + 3 = 2 ∗ 3 (\displaystyle 3+3=2*3). Here are other examples:

   When multiplying powers with the same base, their exponents are added (the base does not change). For example, given the expression x 2 ∗ x 5 (\displaystyle x^(2)*x^(5)). In this case, you just need to add the indicators, leaving the base unchanged. Thus, x 2 ∗ x 5 = x 7 (\displaystyle x^(2)*x^(5)=x^(7)). Here is a visual explanation of this rule:

   When raising a power to a power, the exponents are multiplied. For example, given a degree (x 2) 5 (\displaystyle (x^(2))^(5)). Since exponents are multiplied, then (x 2) 5 = x 2 ∗ 5 = x 10 (\displaystyle (x^(2))^(5)=x^(2*5)=x^(10)). The point of this rule is that you multiply by powers (x 2) (\displaystyle (x^(2))) on itself five times. Like this:

   A power with a negative exponent should be converted to a fraction (reverse power). It doesn't matter if you don't know what a reciprocal degree is. If you are given a degree with a negative exponent, e.g. 3 − 2 (\displaystyle 3^(-2)), write this degree in the denominator of the fraction (put 1 in the numerator), and make the exponent positive. In our example: 1 3 2 (\displaystyle (\frac (1)(3^(2)))). Here are other examples:

   When dividing degrees with the same base, their exponents are subtracted (the base does not change). The division operation is the opposite of the multiplication operation. For example, given the expression 4 4 4 2 (\displaystyle (\frac (4^(4))(4^(2)))). Subtract the exponent in the denominator from the exponent in the numerator (do not change the base). Thus, 4 4 4 2 = 4 4 − 2 = 4 2 (\displaystyle (\frac (4^(4))(4^(2)))=4^(4-2)=4^(2)) = 16 .

   Below are some expressions that will help you learn to solve problems with exponents. The expressions given cover the material presented in this section. To see the answer, simply select the empty space after the equal sign.

Solving problems with fractional exponents

Degree with a fractional indicator (for example, x 1 2 (\displaystyle x^(\frac (1)(2))) ) is converted into a root extraction operation. In our example: x 1 2 (\displaystyle x^(\frac (1)(2))) = x (\displaystyle (\sqrt (x))). Here it does not matter what number is in the denominator of the fractional exponent. For example, x 1 4 (\displaystyle x^(\frac (1)(4)))- is the fourth root of “x”, that is x 4 (\displaystyle (\sqrt[(4)](x))) .