## What is a power in math?

A **power** is the product of several identical factors. For example:

2 Γ 2 Γ 2

The value of this expression is 8

2 Γ 2 Γ 2 = 8

The left part of this equation can be made shorter - first write the repeating multiplier and specify over it how many times it repeats. The repeating factor in this case is 2. It is repeated three times. So above the 2 we write the 3:

2^{3} = 8

The expression reads *"two to the third power equals eight"* or *"the third power of 2 equals 8"*.

The short form of recording multiplication of identical factors is used more often. Therefore, remember that if another number is written over some number, it is the multiplication of several identical factors.

For example, if an expression 5^{3} is given, keep in mind that this expression is equivalent to writing 5 Γ 5 Γ 5.

The number that is repeated is called the **base of the power**. In expression 5^{3}, the base of the power is number 5.

The number written above number 5 is called the **exponent**. In expression 5^{3}, the exponent is number 3. The exponent shows how many times the base of the power is repeated. In this case, base 5 is repeated three times

The operation of multiplying identical factors is called **exponentiation**.

For example, if you want to find the product of four identical factors, each of which is equal to 2, they say that the number 2 is **raised to the fourth power**:

We see that the number 2 to the fourth power is number 16.

Note that in this lesson we are looking at **powers with a natural exponent**. This is a type of power whose exponent is a natural number. Recall that natural numbers are integers that are greater than zero. For example, 1, 2, 3, and so on.

In general, the definition of a power with a natural exponent is as follows:

**The power of a with a natural exponent n is an expression of the form a^{n}, which is equal to the product of n factors, each of which is equal to a**

**examples:**

One must be careful when taking a number to the power. Often a person is inattentive and multiplies the base of a power by the exponent.

For example, the number 5 to the second power is the product of two factors, each of which is 5. This product is 25

Now suppose we inadvertently multiplied base 5 by 2.

There is an error because the number 5 to the second power is not equal to 10.

Additionally, it should be mentioned that the power of a number with exponent 1 is the number itself:

For example, the number 5 to the first power is the number 5 itself

Accordingly, if a number has no exponent, then we must assume that the exponent is equal to one.

For example, numbers 1, 2, 3 are given without exponent, so their exponents will be equal to one. Each of these numbers can be written with exponent 1

And if you exponentiate 0, you get 0. Indeed, no matter how many times nothing is multiplied by itself you get nothing. Examples:

And expression 0^{0} has no meaning. But in some branches of mathematics, in particular analysis and set theory, expression 0^{0} may make sense.

To practice, let's solve some examples on raising numbers to a power.

**Example 1.** To raise number 3 to the second power.

Number 3 to the second power is the product of two factors, each of which is equal to 3

3^{2} = 3 Γ 3 = 9

**Example 2.** To raise the number 2 to the fourth power.

The number 2 to the fourth power is the product of four factors, each of which is 2

2^{4} =2 Γ 2 Γ 2 Γ 2 = 16

**Example 3.** Raise number 2 to the third power.

The number 2 to the third power is the product of three factors, each of which is 2

2^{3} =2 Γ 2 Γ 2 = 8

## Powers of 10

To raise a number 10 to a power, it is enough to add after one a number of zeros equal to the exponent.

For example, let's raise the number 10 to the second power. First write down the number 10 itself and as an exponent write down the number 2

10^{2}

Now put an equal sign, write one and after this one write two zeros, because the number of zeros must be equal to the exponent

10^{2} = 100

So the number 10 to the second power is the number 100. This is because the number 10 to the second power is the product of two factors, each of which is equal to 10

10^{2} = 10 Γ 10 = 100

**Example 2.** Let's raise the number 10 to the third power.

In this case there will be three zeros after the one:

10^{3} = 1000

**Example 3.** Let's raise the number 10 to the fourth power.

In this case there will be four zeros after the one:

10^{4} = 10000

**Example 4.** Let's raise the number 10 to the first power.

In this case there will be one zero after the one:

10^{1} = 10

## Numbers 10, 100, 1000 as a power with base 10

To represent the numbers 10, 100, 1000 and 10000 as a power with base 10, you need to write the base of 10, and as the exponent specify a number equal to the number of zeros of the original number.

Let's represent the number 10 as a power with base 10. We see that it has one zero. So the number 10 as a power with base 10 will be represented as 10^{1}

10 = 10^{1}

**Example 2.** Let us represent the number 100 as a power of base 10. We see that number 100 contains two zeros. Therefore, the number 100 as a power with base 10 is represented as 10^{2}

100 = 10^{2}

**Example 3.** Let's represent the number 1,000 as a power with base 10.

1 000 = 10^{3}

**Example 4.** Let's represent the number 10,000 as a power with base 10.

10 000 = 10^{4}

## Negative numbers and exponents

If you raise a negative number to a power, you must put it in brackets.

For example, let's raise a negative number -2 to the second power. The number -2 to the second power is the product of two factors, each of which is equal to (-2)

(β2)^{2} = (β2) Γ (β2) = 4

If we didn't bracket the number -2, we would be calculating the expression β2^{2}, which is not equal to 4. The expression -2Β² would be -4. To understand why, let's touch on a few points.

When we put a minus in front of a positive number, we thereby perform the **operation of taking the opposite value**.

Suppose a number 2 is given and we need to find its opposite number. We know that the opposite of 2 is -2. To find the opposite of 2, we just need to put minus in front of the number. Inserting a minus in front of a number is already considered a full-fledged operation in mathematics. This operation, as mentioned above, is called the operation of taking the opposite value.

In the case of expression β2^{2} there are two operations: the operation of taking the opposite value and raising to a power. Exponentiation is a higher priority operation than taking the opposite value.

Therefore, expression β2^{2} is calculated in two steps. First, an exponentiation operation is performed. In this case, the positive number 2 was raised to the second power

Then the opposite value was taken. This opposite value was found for the value 4. And the opposite value for 4 is -4

β2^{2 } = β4

The brackets, on the other hand, have the highest execution priority. Therefore, in the case of calculating the expression (β2)^{2}, we first take the opposite value and then raise the negative number -2 to the second power. The result is positive 4, because the product of negative numbers is a positive number.

**Example 2.** Raise -2 to the third power.

-2 to the third power is the product of three factors, each of which is equal to (-2)

(β2)^{3} = (β2) Γ (β2) Γ (β2) = β8

**Example 3.** Raise -2 to the fourth power.

The number -2 to the fourth power is the product of four factors, each of which is equal to (-2)

(β2)^{4} = (β2) Γ (β2) Γ (β2) Γ (β2) = 16

It is easy to see that raising a negative number to a power can result in either a positive or a negative answer. The sign of the answer depends on the exponent of the original power.

If the exponent is even, the answer is positive. If the exponent is odd, the answer will be negative. Let's show this using the number -3 as an example

In the first and third cases, the exponent was an **odd** number, so the answer became **negative**.

In the second and fourth cases, the exponent was an **even** number, so the answer became **positive**.

**Example 7.** Raise the number -5 to the third power.

The number -5 to the third power is the product of three factors each equal to -5. The exponent 3 is an odd number, so we can say in advance that the answer will be negative:

(β5)^{3} = (β5) Γ (β5) Γ (β5) = β125

**Example 8.** Raise the number -4 to the fourth power.

The number -4 to the fourth power is the product of four factors, each of which is -4. The exponent of 4 is even, so we can say in advance that the answer will be positive:

(β4)^{4} = (β4) Γ (β4) Γ (β4) Γ (β4) = 256

## Finding values of expressions

When finding values of expressions that do not contain brackets, the exponentiation will be performed first, followed by multiplication and division in order, and then addition and subtraction in order.

**Example 9.** Find the value of the expression 2 + 5^{2}

First we perform exponentiation. In this case, number 5 is raised to the second power to obtain 25. Then this result is added to the number 2

2 + 5^{2} = 2 + 25 = 27

**Example 10.** Find the value of the expression β6^{2} Γ (β12)

First we perform exponentiation. Note that the number -6 is not bracketed, so number 6 will be raised to the second power, and then the result will be preceded by minus:

β6^{2} Γ (β12) = β36 Γ (β12)

Complete the example by multiplying -36 by (-12)

β6^{2} Γ (β12) = β36 Γ (β12) = 432

**Example 11.** Find the value of the expression β3 Γ 2^{2}

First we perform exponentiation. Then the result is multiplied with the number -3

β3 Γ 2^{2} = β3 Γ 4 = β12

If the expression contains parentheses, you must first perform the actions in those brackets, then exponentiation, then multiplication and division, and then addition and subtraction.

**Example 12.** Find the value of the expression (3^{2} + 1 Γ 3) β 15 + 5

First perform the actions in brackets. Inside the parentheses, we apply the rules we learned earlier, i.e. first raise number 3 to the second power, then multiply 1 Γ 3, then add the results of raising number 3 to the power and multiplying 1 Γ 3. Subtraction and addition are then performed in order. Let's arrange this order of operations over the original expression:

(3^{2} + 1 Γ 3) β 15 + 5 = 12 β 15 + 5 = 2

**Example 13.** Find the value of the expression 2 Γ 5^{3} + 5 Γ 2^{3}

First, let's raise the numbers to a power, then perform multiplication and add up the results:

2 Γ 5^{3} + 5 Γ 2^{3} = 2 Γ 125 + 5 Γ 8 = 250 + 40 = 290

## Identical transformations of powers

Various identity transformations can be performed on powers, thereby simplifying them.

Suppose we need to calculate the expression (2^{3})^{2}. In this example, two to the third power is raised to the second power. In other words, the power is raised to another power.

(2^{3})^{2} is the product of two powers, each of which is 2^{3}

Each of these powers is the product of three factors, each of which is equal to 2

We obtain the product 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2, which is 64. Then the value of the expression (2^{3})^{2} or equals 64

This example can be greatly simplified. To do this, the indices of expression (2^{3})^{2} can be multiplied and the product can be written over base 2

We got 2^{6}. Two to the sixth power is the product of six factors, each of which is 2. This product is 64

This property works because 2^{3} is the product of 2 Γ 2 Γ 2, which in turn is repeated twice. Then it turns out that the base 2 is repeated six times. Hence we can write that 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 is 2^{6}

In general, for any basis **a** with exponent **m** and **n**, the following equality holds:

(*a ^{n}*)

^{m}= a^{n Γ m}This identity transformation is called **raising the power to a power**. It can be read as follows: *"When raising the power to a power, the base is left unchanged and the exponents are multiplied"*.

After multiplying the exponents, you get another power, the value of which can be found.

**Example 2.** Find the value of the expression (3^{2})^{2}

In this example, the base is 3 and the numbers 2 and 2 are exponents. Let's use the rule of raising to a power. Leave the base unchanged, and multiply the exponents:

We got 3^{4}. And the number 3 to the fourth power is 81

Consider the rest of the transformations.

## Multiplying Powers

To multiply the powers, you need to calculate each power separately, and multiply the results.

For example, multiply 2^{2} by 3^{3}.

2^{2} is number 4 and 3^{3} is number 27. Multiply the numbers 4 and 27 and you get 108

2^{2} Γ 3^{3} = 4 Γ 27 = 108

In this example, the bases of the powers were different. In case the bases are the same, you can write one base, and as the exponent write the sum of the exponents of the original powers.

For example, multiply 2^{2} by 2^{3}

In this example, the bases of the powers are the same. In this case we can write down one base 2 and as an exponent write the sum of exponents of powers 2^{2} and 2^{3}, i.e. we can leave the base unchanged, and add the exponents of the original powers. It will look like this:

We got 2^{5}. The number 2 to the fifth power is 32

This property works because 2^{2} is the product of 2 Γ 2, and 2^{3} is the product of 2 Γ 2 Γ 2. Then we obtain the product of five identical factors, each of which is equal to 2. This product can be represented as 2^{5}

In general, for any **a** and the exponents **m** and **n**, the following equality holds:

This identity transformation is called the basic property of the power. It can be read as follows: *"When multiplying powers with equal bases, the base is left unchanged and the exponents are added"*.

Note that this transformation can be applied to any number of powers. The main thing is that the basis is the same.

For example, find the value of the expression 2^{1} Γ 2^{2} Γ 2^{3}. Let's leave the base 2 unchanged, and add up the exponents:

In some tasks, it is sufficient to perform the corresponding transformation without calculating the final power. This is, of course, very convenient, since calculating large powers is not easy.

**Example 1.** Present the expression 5^{8} Γ 25 as a power

In this task we need to make one power instead of the expression 5^{8} Γ 25

The number 25 can be represented as 5^{2}. Then we get the following expression:

In this expression we can apply the main property of the power - leave the base 5 unchanged, and add the exponents 8 and 2:

The task can be considered solved because we have presented the expression 5^{8} Γ 25 as one power, namely as the power of 5^{10}.

Let us write down the solution briefly:

**Example 2.** Present the expression 2^{9} Γ 32 as a power

The number 32 can be represented as 2^{5}. Then we get the expression 2^{9} Γ 2^{5}. Next, we can apply the base property of the power - leave base 2 unchanged, and add the exponents 9 and 5. The result is the following solution:

**Example 3.** Calculate the product 3 Γ 3 using the basic property of a power.

Everyone knows very well that three times three equals nine, but the task requires you to use the basic property of the power in the solution. How do we do this?

Recall that if the number is given without an exponent, then the exponent must be considered equal to one. Therefore the factors 3 and 3 can be written as 3^{1} and 3^{1}

3^{1} Γ 3^{1}

Now let's use the basic property of the power. We leave the base 3 unchanged, and add 1 and 1:

3^{1} Γ 3^{1} = 3^{2}

Then we calculate the value of the expression. The number 3 to the second power is equal to 9

3^{1} Γ 3^{1} = 3^{2} = 9

**Example 4.** Calculate the product 2 Γ 2 Γ 3^{2} Γ 3^{3} using the basic property of powers.

Replace the product of 2 Γ 2 by 2^{1} Γ 2^{1}, then by 2^{1 + 1}, and then by 2^{2}. Replace the product of 3^{2} Γ 3^{3} by 3^{2 + 3} and then by 3^{5}

Then we calculate the value of each power and find the product:

**Example 5.** Perform the multiplication of x Γ x

These are two identical alphabetic factors with exponents of 1. For clarity, write down the exponents. Next, leave the base of x unchanged and add the exponents:

When answering a task like this at school, you should not write down the multiplication of powers with equal bases in such detail as is done here. Such calculations should be done in your head. The teacher is likely to be annoyed and give you a lower grade. Here, however, detailed notation is given so that the material will be as easy to understand as possible.

The solution to this example should preferably be written like this:

**Example 6.** Multiply *x*^{2}* Γ x*

The exponent of the second factor is one. Let us write it down for clarity. Next, leave the base unchanged, and add the exponents:

**Example 7.** Multiply *y*^{3}* Γ y^{2} Γ y*

The exponent of the third factor is one. Let us write it down for clarity. Next, leave the base unchanged, and add the exponents:

**Example 8.** Multiply a*Γ a^{3}Γa^{2}Γa^{5}*

The exponent of the first factor is one. Let us write it down for clarity. Next, leave the base unchanged, and add the exponents:

**Example 9.** Present power 3^{8} as a product of powers with equal bases.

In this task we have to make a product of powers whose bases will be 3 and whose sum of exponents will be 8. Any exponent can be used. Let us represent power 3^{8} as a product of powers 3^{5} and 3^{3}

In this example we again relied on the basic property of the power. After all, the expression 3^{5} Γ 3^{3} can be written as 3^{5 + 3}, hence 3^{8}.

Of course, it was possible to represent power 3^{8} as a product of other powers. For example, as 3^{7} Γ 3^{1}, since this product also equals 3^{8}

Representing a power as a product of powers with identical bases is mostly creative work. So you don't have to be afraid to experiment.

**Example 10.** Represent the power of *x*^{12} as different products of powers with bases x.

Let us use the basic property of power. Let's represent *x*^{12} as products with bases x, and sum of exponents equal to 12

The constructions with the sums of the exponents were written for clarity. Most often they can be omitted. Then you get a compact solution:

## Power of a Product

In order to raise a product to a power, you must raise each factor of that product to a specified power and multiply the results.

For example, raise to the second power the product 2 Γ 3. Put this product in brackets and give 2 as the exponent

Now raise each factor of the product 2 Γ 3 to the second power and multiply the results:

The way this rule works is based on the definition of power, which was given at the beginning.

To raise the product 2 Γ 3 to the second power is to repeat the product twice. And if you repeat it twice, you can get the following:

2 Γ 3 Γ 2 Γ 3

The product does not change from rearranging the places of the factors. This allows you to group the same factors together:

2 Γ 2 Γ 3 Γ 3

Repeating factors can be replaced by short entries - bases with exponents. The product 2 Γ 2 can be replaced by 2^{2}, and the product 3 Γ 3 can be replaced by 3^{2}. Then the expression 2 Γ 2 Γ 3 Γ 3 turns into 2^{2} Γ 3^{2}.

Let **ab** be the original product. To raise this product to power **n**, we must separately raise the factors **a** and **b** to the specified power **n**

This property holds for any number of factors. The following expressions are also valid:

**Example 2.** Find the value of the expression (2 Γ 3 Γ 4)^{2}

In this example, the product 2 Γ 3 Γ 4 must be raised to the second power. To do this, take each factor of the product to the second power and multiply the results:

**Example 3.** To raise the product *a Γ b Γ c* to the third power

Put this product in brackets, and as an exponent we give the number 3

Next, raise each factor of the product to the third power:

**example 4**. Raise to the third power the product 3 Γ x Γ y Γ z

Let's bracket this product, and as an exponent let's specify 3

(3 Γ *x Γ y Γ z*)^{3}

Let us raise each factor of the product to the third power:

(3 Γ *x Γ y Γ z*)^{3} = 3^{3}*x*^{3}*y*^{3}*z*^{3}

The number 3 to the third power is 27. Let's leave the rest unchanged:

(3*xyz*)^{3} = 3^{3}*x*^{3}*y*^{3}*z*^{3} = 27*x*^{3}*y*^{3}*z*^{3}

In some examples, multiplication of powers with equal exponent can be replaced by the product of bases with one exponent.

For example, calculate the value of the expression 5^{2} Γ 3^{2}. Raise each number to the second power and multiply the results:

5^{2} Γ 3^{2} = 25 Γ 9 = 225

But it is not necessary to calculate each power separately. Instead, the given product of powers can be replaced by a product with one exponent (5 Γ 3)^{2}. Then calculate the value in brackets and raise the result to the second power:

5^{2} Γ 3^{2} = (5 Γ 3)^{2} = (15)^{2} = 225

In this case again the rule of exponentiation of the product was used. If (*a Γ b*)* ^{n}* =

*a*, then

^{n}Γ b^{n}*a*

^{n}Γ b^{n}*. That is, the left and right parts of the equality have swapped places.*

^{ }= (a Γ b)^{n}## Power of a Power Rule (Exponents)

We considered this transformation as an example when we tried to understand the essence of identical power transformations.

The base is left unchanged and the exponents are multiplied when the power is raised to a power:

(*a ^{n}*)

^{m}= a^{n Γ m}For example, the expression (2^{3})^{2} is an increase to power - two to the third power is raised to the second power. To find the value of this expression, the base can be left unchanged, and the exponents multiplied:

(2^{3})^{2} = 2^{3 Γ 2} = 2^{6}

Then calculate the power of 2^{6}, which is 64

(2^{3})^{2} = 2^{3 Γ 2} = 2^{6} = 64

This rule is based on the previous rules: exponentiation of the product and the main property of the power.

Let us return to the expression (2^{3})^{2}. The expression in brackets 2^{3} is the product of three identical factors, each equal to 2. Then in expression (2^{3})^{2} the power inside the brackets can be replaced by the product 2 Γ 2 Γ 2.

(2 Γ 2 Γ 2)^{2}

And this is the exponentiation of the product that we studied earlier. Recall that to exponentiate a product, each factor of the product must be multiplied to the specified power and the results multiplied:

(2 Γ 2 Γ 2)^{2} = 2^{2} Γ 2^{2} Γ 2^{2}

Now we are dealing with the main property of the power. We leave the base unchanged, and add the exponents:

(2 Γ 2 Γ 2)^{2} = 2^{2} Γ 2^{2} Γ 2^{2} = 2^{2 + 2 + 2} = 2^{6}

As before we got 2^{6}. The value of this power is 64

(2 Γ 2 Γ 2)^{2} = 2^{2} Γ 2^{2} Γ 2^{2} = 2^{2 + 2 + 2} = 2^{6} = 64

A product whose factors are also powers can also be raised to a power.

For example, find the value of the expression (2^{2} Γ 3^{2})^{3}. Here, the exponents of each factor must be multiplied by the total exponent of 3. Then find the value of each power and calculate the product:

(2^{2} Γ 3^{2})^{3 }= 2^{2Γ3 } Γ 3^{2Γ3} = 2^{6 }Γ 3^{6} = 64 Γ 729 = 46656

The same thing happens when you raise a product to the power of a product. We said that in the power of a product, each factor of that product is raised to a specified power.

For example, to raise the product 2 Γ 4 to the third power, you must write the following expression:

But earlier it was said that if a number is given without an exponent, then the exponent should be considered equal to one. It turns out that the factors of the product 2 Γ 4 originally have exponent equal to 1. It means that the expression 2^{1} Γ 4^{1} was raised to the third power. This is power of a power.

Let's rewrite the solution using the rule of power of a power. We should get the same result:

**Example 2.** Find the value of the expression (3^{3})^{2}

We leave the base unchanged, and multiply the exponents:

We got 3^{6}. The number 3 to the sixth power is 729

**Example 3.** Perform exponentiation of the expression (xy)Β³

Let's raise each factor of the product to the third power:

**Example 4.** Perform exponentiation of the expression (abc)β΅

Let's raise each factor of the product to the fifth power:

**Example 5.** Perform exponentiation of the expression (β2*ax*)^{3}

Let's raise each factor of the product to the third power:

Since the negative number -2 was raised to the third power, it was bracketed.

Next you need to calculate what is being calculated. In this case you can calculate (β2)^{3} to get -8. The letter part will remain unchanged:

**Example 6.** Perform exponentiation of the expression (10*xy*)^{2}

**Example 7.** Perform exponentiation of the expression (β5*x*)^{3}

**Example 8.** Perform exponentiation of the expression (β3*y*)^{4}

**Example 9.** Perform exponentiation of the expression (β2*abx*)β΄

**Example 10.** Simplify the expression *x*^{5} Γ (*x*^{2})^{3}

Let's leave the power of *x*^{5} unchanged for now, and in the expression (*x*^{2})^{3} let's raise the power to a power:

*x*^{5}* Γ *(*x*^{2})^{3}* = x*^{5}* Γ x*^{2 Γ 3}* = x*^{5}* Γ x*^{6}

Now let's multiply *x*^{5}*Γ x*^{6}. To do this, use the basic property of the power - leave the base of **x** unchanged, and add the exponents:

*x*^{5}* Γ *(*x*^{2})^{3}* = x*^{5}* Γ x*^{2Γ 3}* = x*^{5}* Γ x*^{6} = *x*^{5 + 6} = *x*^{11}

**Example 11.** Find the value of the expression 4^{3} Γ 2^{2} using the main property of the power.

The basic property of powers can be used if the bases of the original powers are the same. In this example, the bases are different, so first we have to modify the original expression a bit, namely, to make the bases of the powers to be the same.

Let's take a close look at the power of 4^{3}. The base of this power is number 4, which can be represented as 2^{2}. Then the original expression will look like (2^{2})^{3} Γ 2^{2}. If we raise the power of (2^{2})^{3} to a power, we obtain 2^{6}. Then the original expression will take the form 2^{6} Γ 2^{2}, which can be calculated using the main property of the power.

Let us write down the solution of this example:

## Dividing Power

To do division of powers, you need to find the value of each power, then do division of prime numbers.

For example, divide 4^{3} by 2^{2}.

Calculate 4^{3}, we get 64. Calculate 2^{2}, we get 4. Now divide 64 by 4, we get 16

If the bases are the same when dividing the powers, the base can be left unchanged, and the exponent of the divisor can be subtracted from the exponent of the dividend.

For example, find the value of the expression 2^{3} : 2^{2}

Let's leave the base 2 unchanged, and subtract the exponent of the divisor from the exponent of the dividend:

So the value of the expression 2^{3} : 2^{2} is 2.

This property is based on the multiplication of powers with equal bases, or as we used to say, on the basic property of the power.

Let's go back to the previous example 2^{3} : 2^{2}. Here the dividend is 2^{3} and the divisor is 2^{2}.

To divide one number by another means to find a number that, when multiplied by the divisor, yields a resulting dividend.

In our case, to divide 2^{3} by 2^{2} is to find a power that, when multiplied by the divisor of 2^{2}, results in 2^{3}. And what power can be multiplied by 2^{2} to get 2^{3} ? Obviously, only the power of2^{1}. From the basic property of powers we have:

To make sure that the value of 2^{3} : 2^{2} is 2^{1}, we can directly calculate the expression 2^{3} : 2^{2}. To do this, first find the value of the power of 2^{3}, we get 8. Then we find the value of the power of 2^{2}, we get 4. Divide 8 by 4 to get 2 or 2^{1}, because 2 = 2^{1}.

2^{3} : 2^{2} = 8 : 4 = 2

Thus, when dividing powers with equal bases, the following equality is satisfied:

It may also happen that not only the bases are the same, but also the exponents. In this case the answer will be one.

For example, find the value of the expression 2^{2} : 2^{2}. Calculate the value of each power and divide the resulting numbers:

When solving example 2^{2} : 2^{2}, you can also apply the rule of division of powers with equal bases. The result is a number to the power of zero, because the difference between the exponents of powers 2^{2} and 2^{2} is zero:

In mathematics, it is accepted that any number to the power of zero is one:

We found out above why the number 2 to the power of zero equals one. If you calculate 22 : 22 by the usual method, without using the rule of division of powers, you will get one.

**Example 2.** Find the value of the expression 4^{12} : 4^{10}

Use the rule of division of powers. Let's leave the base 4 unchanged, and subtract the exponent of the divisor from the exponent of the dividend:

4^{12} : 4^{10} = 4^{12 β 10} = 4^{2} = 16

**Example 3.** Present the quotient *x*^{3} : x as a power with base x

Use the rule of division of powers. We leave the base of x unchanged, and subtract the exponent of the divisor from the exponent of the dividend. The exponent of the divisor is one. For clarity, write it down:

**Example 4.** Present the quotient *x*^{3} : *x*^{2} as a power with base x

Let us use the rule of division of powers. Let's leave the base of x unchanged, and subtract the exponent of the divisor from the exponent of the dividend:

The division of powers can be written as a fraction. Thus, the previous example can be written as follows:

The numerator and denominator of a fraction may be written in expanded form, namely, as products of identical factors. The power of *x*^{3} can be written as **x Γ x Γ x**, and the power of *x*^{2} as **x Γ x**. Then the construction *x*^{3 β 2} can be skipped and the fraction can be shortened. The numerator and denominator can be reduced by two x factors each. The result will be one factor x

Or even shorter:

Also, it is useful to be able to quickly reduce fractions consisting of powers. For example, a fraction can be reduced by *x*^{2}. To reduce by *x*^{2}, divide the numerator and denominator of by *x*^{2}

The division of powers may not be described in detail. The given reduction can be made shorter:

Or even shorter:

**Example 5.** Perform division *x*^{12}* : x*^{3}

Let us use the rule of division of powers. Let's leave the base of x unchanged, and subtract the exponent of the divisor from the exponent of the dividend:

Write down the solution using fraction reduction. Let us write the division of powers *x*^{12}* : x*^{3} as . Then let us reduce this fraction by *x*^{3}.

**Example 6.** Find the value of the expression

In the numerator, multiply powers with equal bases:

Now apply the rule of division of powers with equal bases. We leave the base 7 unchanged, and subtract the exponent of the divisor from the exponent of the dividend:

Complete the example by calculating the power of 7^{2}

**Example 7.** Find the value of the expression

In the numerator, raise the power to a power. Do this with expression (2^{3})^{4}

Now do the multiplication of powers with equal bases in the numerator:

Now apply the rule of division of powers with equal bases:

So the value of expression is 16

In some examples, it is possible to reduce the same factors in the course of the solution. This simplifies the expression and the calculation as a whole.

For example, find the value of expression . Let us write the power of 4^{3} as the power of (2^{2})^{3}. Then we obtain the following expression:

In the numerator, perform the exponentiation of the power. Do this with expression (2^{2})^{3}

The numerator and denominator of the resulting expression contain the power 2^{6}, which can be reduced by 2^{6}

We see that this leaves us with the only power of 3^{2}, the value of which is 9.

**Example 8.** Find the value of the expression

The denominator is the product of powers with the same exponent. According to the rule of exponentiation of the product, the construction 7^{5} Γ 4^{5} can be represented as a power with one exponent (7 Γ 4)^{5}. Next, multiply the expression in brackets to get 28^{5}. As a result, the original expression will take the following form:

Now you can apply the rule of division of powers:

So, the value of expression is 28. Let's write down the solution in full:

## Exponents of Fractions

To raise a fraction to a power, you must raise the numerator and denominator of the fraction to a specified power.

For example, raise fraction to the second power. Put this fraction in brackets and give 2 as the exponent

If we do not bracket the whole fraction, it is equivalent to raising only the numerator of the fraction. If we want to raise the fraction to the second power, we should not write it as .

So, to calculate the value of expression , you need to raise the numerator and denominator of this fraction to the second power:

We obtained a fraction with powers in the numerator and denominator. Calculate each power separately

So the fraction to the second power is equal to the fraction .

The above rule works as follows. The fraction to the second power is the product of two fractions, each of which is equal to

We remember that to multiply fractions, you must multiply their numerators and denominators:

And since the numerator and denominator are multiplied by the same factors, the expressions 2 Γ 2 and 3 Γ 3 can be replaced by 2^{2} and 3^{2}, respectively:

From which you get the answer .

In general, for any **a** and **b** **β 0** the following equality holds:

This identity transformation is called **exponentiation of a fraction**.

**Example 2.** To raise fraction to the third power

We conclude this fraction in brackets and give number 3 as the exponent. Then raise the numerator and denominator of the fraction to the third power and calculate the resulting fraction:

A negative fraction is raised to a power in the same way, but before calculating you should decide what sign the answer will have. If the exponent is even, the answer will be positive. If the exponent is odd, the answer will be negative.

For example, raise fraction to the second power:

The exponent is an even number. So the answer will be positive. Then we apply the rule of raising a fraction to a power and calculate the resulting fraction:

The answer is positive because is the product of two factors, each equal to a fraction of

And the product of negative numbers (including rational numbers) is a positive number:

If the fraction is raised to the third power, the answer will be negative, because in this case the exponent will be an odd number. The rule of exponentiation remains the same, but before performing this exponentiation, we will need to put a minus:

Here the answer is negative because expression is the product of three factors, each of which is a fraction of .

First we multiplied and and got , but then we multiplied by and got a negative answer of

**Example 3.** Find the value of the expression

Let's perform exponentiation of fractions:

Then we calculate the value of the resulting expression:

## Exponents of Decimals

When you raise a decimal to the power, you must put it in brackets. For example, let's raise a decimal 1.5 to the second power

It is allowed to convert a decimal to a fraction and to exponentiate that fraction. Solve the previous example by converting a decimal to a fraction:

**Example 2.** Find the value of the power of (β1.5)^{3}

The exponent is an odd number. So the answer will be negative

**Example 3.** Find the value of the power of (β2.4)^{2}

The exponent is an even number. So the answer will be positive:

2. If you find an error or inaccuracy, please describe it.

3. Positive feedback is welcome.