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Lesson Explainer: Laws of Exponents Mathematics • 8th Grade

In this explainer, we will learn how to apply the laws of exponents to multiply and divide powers and work out a power raised to a power.

We begin by recalling that repeated multiplication is represented by exponentiation. In general, we say that raising a number to a positive integer exponent ๐‘› is the same as multiplying it by itself such that it appears ๐‘› times in the product. If we have a base of ๐‘Ž๐‘โˆˆโ„š, then we have the following: ๏€ป๐‘Ž๐‘๏‡=๏‡„๏†ช๏†ช๏†ช๏‡…๏†ช๏†ช๏†ช๏‡†๐‘Ž๐‘ร—๐‘Ž๐‘ร—โ‹ฏร—๐‘Ž๐‘=๐‘Žร—๐‘Žร—โ‹ฏร—๐‘Ž๐‘ร—๐‘ร—โ‹ฏร—๐‘=๐‘Ž๐‘.๏Š๏Š๏Š๏Štimes

This definition and result allow us to prove some useful results in evaluating exponential expressions. We call these the laws of exponents.

First, we can consider multiplying two exponential expressions with the same base. For example, 2ร—2๏Šฉ๏Šช. We have 2ร—2=๏‡„๏†ช๏†ช๏‡…๏†ช๏†ช๏‡†(2ร—2ร—2)ร—๏‡„๏†ช๏†ช๏†ช๏‡…๏†ช๏†ช๏†ช๏‡†(2ร—2ร—2ร—2)=๏‡„๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏‡…๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏‡†2ร—2ร—2ร—2ร—2ร—2ร—2=2.๏Šฉ๏Šช๏Šฉ๏Šช๏Šฉ๏Šฐ๏Šช(๏Šฉ๏Šฐ๏Šช)timestimestimes

This suggests that when we multiply exponential expressions with the same base, we add the exponents. We can show that this result holds in general by following the same process.

Letโ€™s say that ๐‘Ž๐‘โˆˆโ„š and ๐‘› and ๐‘š are positive integers; then, ๏€ป๐‘Ž๐‘๏‡ร—๏€ป๐‘Ž๐‘๏‡=๏‡„๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏‡…๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏‡†๏€ป๏€ป๐‘Ž๐‘๏‡ร—๏€ป๐‘Ž๐‘๏‡ร—โ‹ฏร—๏€ป๐‘Ž๐‘๏‡๏‡ร—๏‡„๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏‡…๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏‡†๏€ป๏€ป๐‘Ž๐‘๏‡ร—๏€ป๐‘Ž๐‘๏‡ร—โ‹ฏร—๏€ป๐‘Ž๐‘๏‡๏‡.๏Š๏‰๏Š๏‰timestimes

Remember we can evaluate a product of rational numbers in any order, so we can rewrite this as a product of ๐‘Ž๐‘ with itself where it appears ๐‘›+๐‘š times. Hence, ๏€ป๐‘Ž๐‘๏‡ร—๏€ป๐‘Ž๐‘๏‡=๏‡„๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏‡…๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏‡†๏€ป๏€ป๐‘Ž๐‘๏‡ร—๏€ป๐‘Ž๐‘๏‡ร—โ‹ฏร—๏€ป๐‘Ž๐‘๏‡๏‡ร—๏‡„๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏‡…๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏‡†๏€ป๏€ป๐‘Ž๐‘๏‡ร—๏€ป๐‘Ž๐‘๏‡ร—โ‹ฏร—๏€ป๐‘Ž๐‘๏‡๏‡=๏‡„๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏‡…๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏‡†๏€ป๐‘Ž๐‘๏‡ร—๏€ป๐‘Ž๐‘๏‡ร—โ‹ฏร—๏€ป๐‘Ž๐‘๏‡=๏€ป๐‘Ž๐‘๏‡.๏Š๏‰๏Š๏‰๏Š๏Šฐ๏‰๏Š๏Šฐ๏‰timestimestimes

We can follow a similar process if we are dividing two exponential expressions with the same base. However, we also need the exponent of the dividend to be greater than the exponent of the divisor.

For example, if we want to calculate ๏€ผ23๏ˆรท๏€ผ23๏ˆ๏Šซ๏Šช, we have ๏€ผ23๏ˆรท๏€ผ23๏ˆ=23รท23.๏Šซ๏Šช๏Šซ๏Šซ๏Šช๏Šช

Instead of dividing, we multiply by the reciprocal to get 23รท23=23ร—32.๏Šซ๏Šซ๏Šช๏Šช๏Šซ๏Šซ๏Šช๏Šช

We then write out the exponents in full and cancel out the shared factors:

Since we cancel out four of the factors of 2 and 3 from the powers of 5, we can think of this as ๏€ผ23๏ˆ๏Šซ๏Šฑ๏Šช. This then suggests that when we divide exponential expressions with the same base, we subtract the exponents. We can show that this result holds for any rational number by following the same process.

If ๐‘Ž๐‘โˆˆโ„š and ๐‘› and ๐‘š are positive integers with ๐‘›>๐‘š, we have ๏€ป๐‘Ž๐‘๏‡รท๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘รท๐‘Ž๐‘=๐‘Ž๐‘ร—๐‘๐‘Ž.๏Š๏‰๏Š๏Š๏‰๏‰๏Š๏Š๏‰๏‰

We can rearrange by writing the products out in full to get

Since ๐‘›>๐‘š, we can cancel out ๐‘š of the shared factors of ๐‘Ž in the first fraction and ๐‘š of the shared factors of ๐‘ in the second fraction. This will leave ๐‘›โˆ’๐‘š factors of ๐‘Ž in the numerator and ๐‘›โˆ’๐‘š factors of ๐‘ in the denominator. Hence, ๏€ป๐‘Ž๐‘๏‡รท๏€ป๐‘Ž๐‘๏‡=๏‡„๏†ช๏†ช๏†ช๏†ช๏‡…๏†ช๏†ช๏†ช๏†ช๏‡†(๐‘Žร—๐‘Žร—โ‹ฏร—๐‘Ž)1ร—1(๐‘ร—๐‘ร—โ‹ฏร—๐‘)๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Ž=๐‘Žร—1๐‘=๐‘Ž๐‘.๏Š๏‰๏Š๏Šฑ๏‰๏Š๏Šฑ๏‰๏Š๏Šฑ๏‰๏Š๏Šฑ๏‰๏Š๏Šฑ๏‰๏Š๏Šฑ๏‰factorsfactors

Finally, we can rewrite this as an exponent of ๐‘Ž๐‘. We have ๏€ป๐‘Ž๐‘๏‡รท๏€ป๐‘Ž๐‘๏‡=๏€ป๐‘Ž๐‘๏‡.๏Š๏‰๏Š๏Šฑ๏‰

There is one last rule we can deduce by using this definition. We can consider what would happen if we raised a base to multiple exponents.

For example, letโ€™s evaluate ๏€น4๏…๏Šจ๏Šฉ. We start with the expression inside the parentheses, so ๏€น4๏…=(4ร—4).๏Šจ๏Šฉ๏Šฉ

We then need to cube this expression. So, we get (4ร—4)=(4ร—4)ร—(4ร—4)ร—(4ร—4).๏Šฉ

We see that this is 4 multiplied by itself, where it appears 6 times in the product. However, we can see where this value of 6 comes from. Since we cube 4ร—4, we multiply the number of factors by 3, which gives us 2ร—3=6 factors of 4. Hence, we have ๏€น4๏…=(4ร—4)ร—(4ร—4)ร—(4ร—4)=4.๏Šจ๏Šฉ(๏Šจร—๏Šฉ)

We can show that this result holds for any rational number by following the same process.

If ๐‘Ž๐‘โˆˆโ„š and ๐‘› and ๐‘š are positive integers, then

We can summarize all the results we have just proven as follows.

Rules: Laws of Exponents for Rational Bases

If ๐‘Ž๐‘โˆˆโ„š and ๐‘› and ๐‘š are positive integers, then

  1. ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘๏Š๏Š๏Š,
  2. ๏€ป๐‘Ž๐‘๏‡ร—๏€ป๐‘Ž๐‘๏‡=๏€ป๐‘Ž๐‘๏‡๏Š๏‰๏Š๏Šฐ๏‰,
  3. ๏€ผ๏€ป๐‘Ž๐‘๏‡๏ˆ=๏€ป๐‘Ž๐‘๏‡๏Š๏‰๏Šร—๏‰,
  4. and if ๐‘›>๐‘š, then
    ๏€ป๐‘Ž๐‘๏‡รท๏€ป๐‘Ž๐‘๏‡=๏€ป๐‘Ž๐‘๏‡๏Š๏‰๏Š๏Šฑ๏‰.

Letโ€™s now see some examples of using the laws of exponents to evaluate and simplify exponential expressions.

Example 1: Multiplying Negative Mixed Number Expressions with Positive Integer Exponents

Which of the following is equal to ๏€ผโˆ’112๏ˆร—๏€ผโˆ’112๏ˆ๏Šฉ๏Šจ?

  1. 71932
  2. โˆ’71932
  3. 32243
  4. โˆ’31251024
  5. โˆ’243

Answer

We could evaluate this expression by evaluating each factor separately; however, we can also simplify the expression first by using the laws of exponents. We recall that if we multiply two powers with the same base, then we can just add the exponents, ๐‘ร—๐‘=๐‘,๏Š๏‰๏Š๏Šฐ๏‰ for any rational number ๐‘ and positive integers ๐‘› and ๐‘š.

So, ๏€ผโˆ’112๏ˆร—๏€ผโˆ’112๏ˆ=๏€ผโˆ’112๏ˆ=๏€ผโˆ’112๏ˆ.๏Šฉ๏Šจ๏Šฉ๏Šฐ๏Šจ๏Šซ

We can evaluate this expression by first rewriting โˆ’112 as a fraction. We note that โˆ’112=โˆ’32. Thus, ๏€ผโˆ’112๏ˆ=๏€ผโˆ’32๏ˆ.๏Šซ๏Šซ

Finally, we recall that if ๐‘Ž๐‘ is a rational number and ๐‘› is a positive integer, then ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘๏Š๏Š๏Š.

Hence, ๏€ผโˆ’32๏ˆ=(โˆ’3)2=โˆ’24332.๏Šซ๏Šซ๏Šซ

We could leave our answer like this; however, it is not one of the options. Instead, we rewrite this as a mixed number. We have โˆ’24332=โˆ’32ร—7+1932=โˆ’71932, which is option B.

In our next example, we will use a different one of the laws of exponents to evaluate an expression.

Example 2: Dividing Negative Mixed Number Expressions by Positive Integer Exponents

Which of the following has the same value as ๏€ผโˆ’112๏ˆรท๏€ผโˆ’112๏ˆ๏Šฉ๏Šจ?

  1. 23
  2. โˆ’23
  3. 32
  4. โˆ’32
  5. โˆ’24332

Answer

We could evaluate the dividend and divisor separately and then evaluate the division. However, it is easier to simplify the expression first. We recall that the laws of exponents tell us that when dividing powers with the same base, we subtract the exponents, ๐‘รท๐‘=๐‘,๏Š๏‰๏Š๏Šฑ๏‰ for any rational number ๐‘ and positive integers ๐‘› and ๐‘š with ๐‘›>๐‘š.

Thus, ๏€ผโˆ’112๏ˆรท๏€ผโˆ’112๏ˆ=๏€ผโˆ’112๏ˆ=๏€ผโˆ’112๏ˆ.๏Šฉ๏Šจ๏Šฉ๏Šฑ๏Šจ๏Šง

We then recall that raising a number to an exponent of 1 leaves it unchanged, so ๏€ผโˆ’112๏ˆ=โˆ’112.๏Šง

Finally, we convert our answer into a fraction. We have โˆ’112=โˆ’1ร—2+12=โˆ’32, which is option D.

In our next example, we will determine which of a set of given options is equivalent to a given exponential expression.

Example 3: Evaluating Positive Rational Expressions with Positive Integer Exponents Raised to Positive Integer Exponents

Which of the following expressions has the same value as ๏€พ๏€ผ13๏ˆ๏Š๏Šฉ๏Šจ?

  1. ๏€ผ13๏ˆ๏Šฌ
  2. ๏€ผ13๏ˆ๏Šซ
  3. ๏€ผ13๏ˆ๏Šฏ
  4. 23๏Šฉ
  5. ๏€ผ13๏ˆร—๏€ผ13๏ˆ๏Šฉ๏Šจ

Answer

We could evaluate this expression and all of the options by just evaluating the exponents. However, it is easier to simplify the given expression by using the laws of exponents.

We recall that if we raise a rational base to a positive integer exponent and then raise this to another positive integer exponent, then we can simplify by multiplying the exponents, (๐‘)=๐‘,๏Š๏‰๏Šร—๏‰ for any rational number ๐‘ and positive integers ๐‘› and ๐‘š.

Therefore, ๏€พ๏€ผ13๏ˆ๏Š=๏€ผ13๏ˆ=๏€ผ13๏ˆ.๏Šฉ๏Šจ๏Šฉร—๏Šจ๏Šฌ

We see that this is option A.

In our next example, we will use the laws of exponents to simplify the evaluation of a given exponential expression.

Example 4: Evaluating a Numerical Expression Using the Laws of Exponents

Calculate ๏€ปโˆ’3๏‡ร—๏€ปโˆ’1๏‡๏€ปโˆ’๏‡ร—๏€ปโˆ’๏‡๏Šง๏Šซ๏Šญ๏Šง๏Šจ๏Šฌ๏Šง๏Šฌ๏Šซ๏Šฌ๏Šฉ๏Šจ๏Šช, giving your answer in its simplest form.

Answer

We could evaluate this expression by evaluating each power in turn and then multiplying the fractions. However, this is a difficult process and it would be easy to make mistakes. Instead, we can simplify by using the laws of exponents.

First, we rewrite each mixed number as a fraction. We have โˆ’315=โˆ’155โˆ’15=โˆ’165,โˆ’112=โˆ’22โˆ’12=โˆ’32.

This allows us to rewrite the expression as ๏€ปโˆ’3๏‡ร—๏€ปโˆ’1๏‡๏€ปโˆ’๏‡ร—๏€ปโˆ’๏‡=๏€ปโˆ’๏‡ร—๏€ปโˆ’๏‡๏€ปโˆ’๏‡ร—๏€ปโˆ’๏‡.๏Šง๏Šซ๏Šญ๏Šง๏Šจ๏Šฌ๏Šง๏Šฌ๏Šซ๏Šฌ๏Šฉ๏Šจ๏Šช๏Šง๏Šฌ๏Šซ๏Šญ๏Šฉ๏Šจ๏Šฌ๏Šง๏Šฌ๏Šซ๏Šฌ๏Šฉ๏Šจ๏Šช

We can now see that the expression contains the quotients of the same bases raised to positive integer powers. We can rewrite the expression to highlight taking the quotients of the same bases: ๏€ปโˆ’๏‡ร—๏€ปโˆ’๏‡๏€ปโˆ’๏‡ร—๏€ปโˆ’๏‡=๏€ปโˆ’๏‡๏€ปโˆ’๏‡ร—๏€ปโˆ’๏‡๏€ปโˆ’๏‡.๏Šง๏Šฌ๏Šซ๏Šญ๏Šฉ๏Šจ๏Šฌ๏Šง๏Šฌ๏Šซ๏Šฌ๏Šฉ๏Šจ๏Šช๏Šง๏Šฌ๏Šซ๏Šญ๏Šง๏Šฌ๏Šซ๏Šฌ๏Šฉ๏Šจ๏Šฌ๏Šฉ๏Šจ๏Šช

We now recall that for any rational number ๐‘Ž๐‘ and positive integer exponents ๐‘› and ๐‘š such that ๐‘›>๐‘š, we have ๏€ป๐‘Ž๐‘๏‡รท๏€ป๐‘Ž๐‘๏‡=๏€ป๐‘Ž๐‘๏‡.๏Š๏‰๏Š๏Šฑ๏‰

Thus, ๏€ปโˆ’๏‡๏€ปโˆ’๏‡ร—๏€ปโˆ’๏‡๏€ปโˆ’๏‡=๏€ผโˆ’165๏ˆร—๏€ผโˆ’32๏ˆ=๏€ผโˆ’165๏ˆร—๏€ผโˆ’32๏ˆ.๏Šง๏Šฌ๏Šซ๏Šญ๏Šง๏Šฌ๏Šซ๏Šฌ๏Šฉ๏Šจ๏Šฌ๏Šฉ๏Šจ๏Šช๏Šญ๏Šฑ๏Šฌ๏Šฌ๏Šฑ๏Šช๏Šง๏Šจ

We know that raising a number to first power leaves it unchanged. We can also distribute the exponent of 2 over the fraction by recalling that ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘๏Š๏Š๏Š for any rational number ๐‘Ž๐‘ and positive integer ๐‘›. This allows us to rewrite the expression as follows: ๏€ผโˆ’165๏ˆร—๏€ผโˆ’32๏ˆ=๏€ผโˆ’165๏ˆร—๏€ฟ(โˆ’3)2๏‹.๏Šง๏Šจ๏Šจ๏Šจ

We can now note that (โˆ’3)=9๏Šจ and 2=4๏Šจ, allowing us to rewrite this as ๏€ผโˆ’165๏ˆร—๏€ฟ(โˆ’3)2๏‹=๏€ผโˆ’165๏ˆร—94.๏Šจ๏Šจ

We can now evaluate the product ๏€ผโˆ’165๏ˆร—94=โˆ’16ร—95ร—4=โˆ’365.

In our next example, we will evaluate an algebraic expression using the laws of exponents.

Example 5: Evaluating an Algebraic Expression Using the Laws of Exponents

If ๐‘ฅ=15, ๐‘ฆ=12, and ๐‘ง=25, find the value of ๐‘ฅรท(๐‘ฆ๐‘ง)๏Šช๏Šจ.

Answer

We first substitute the given values into the expression. This gives us ๐‘ฅรท(๐‘ฆ๐‘ง)=๏€ผ15๏ˆรท๏€ผ๏€ผ12๏ˆ๏€ผ25๏ˆ๏ˆ.๏Šช๏Šจ๏Šช๏Šจ

We can simplify this expression by evaluating the product inside the square as follows: ๏€ผ12๏ˆ๏€ผ25๏ˆ=1ร—22ร—5=15.

Thus, ๏€ผ15๏ˆรท๏€ผ๏€ผ12๏ˆ๏€ผ25๏ˆ๏ˆ=๏€ผ15๏ˆรท๏€ผ15๏ˆ.๏Šช๏Šจ๏Šช๏Šจ

We can now note that we are dividing exponential expressions with the same rational base. We can recall that the laws of exponents tell us that when dividing exponential expressions with the same rational base, we subtract the exponents, ๐‘รท๐‘=๐‘,๏Š๏‰๏Š๏Šฑ๏‰ for any rational number ๐‘ and positive integers ๐‘› and ๐‘š with ๐‘›>๐‘š.

Hence, ๏€ผ15๏ˆรท๏€ผ15๏ˆ=๏€ผ15๏ˆ=๏€ผ15๏ˆ.๏Šช๏Šจ๏Šช๏Šฑ๏Šจ๏Šจ

Finally, we evaluate the exponent: ๏€ผ15๏ˆ=15=125.๏Šจ๏Šจ

In our final example, we will simplify an algebraic expression using the laws of exponents.

Example 6: Simplifying an Algebraic Expression Using the Laws of Exponents

Given that x is a rational number, which of the following expressions is equivalent to ๐‘ฅร—๐‘ฅ+๏€น๐‘ฅ๏…๏Šซ๏Šช๏Šช๏Šจ?

  1. ๐‘ฅ+๐‘ฅ๏Šฏ๏Šฎ
  2. ๐‘ฅ+๐‘ฅ๏Šจ๏Šฆ๏Šฎ
  3. ๐‘ฅ+๐‘ฅ๏Šจ๏Šฆ๏Šฌ
  4. ๐‘ฅ+๐‘ฅ๏Šฏ๏Šฌ
  5. ๐‘ฅ๏€น๐‘ฅ+๐‘ฅ๏…๏Šช๏Šซ๏Šจ

Answer

Since ๐‘ฅ is a rational number, we can apply the laws of exponents to simplify this algebraic expression. We note that the expression contains the product of exponential expressions with the same rational base and the repeated exponentiation of a rational base. So, we recall the two laws of exponents ๐‘ร—๐‘=๐‘,(๐‘)=๐‘,๏Š๏‰๏Š๏Šฐ๏‰๏Š๏‰๏Šร—๏‰ for any rational number ๐‘ and positive integers ๐‘› and ๐‘š.

Thus, ๐‘ฅร—๐‘ฅ=๐‘ฅ=๐‘ฅ๏Šซ๏Šช๏Šซ๏Šฐ๏Šช๏Šฏ and ๏€น๐‘ฅ๏…=๐‘ฅ=๐‘ฅ.๏Šช๏Šจ๏Šชร—๏Šจ๏Šฎ

We can then substitute these into the expression to get ๐‘ฅร—๐‘ฅ+๏€น๐‘ฅ๏…=๐‘ฅ+๐‘ฅ,๏Šซ๏Šช๏Šช๏Šจ๏Šฏ๏Šฎ which we can see is option A.

Letโ€™s finish by recapping some of the important points from this explainer.

Key Points

  • If ๐‘Ž๐‘โˆˆโ„š and ๐‘› is a positive integer, then ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘.๏Š๏Š๏Š
  • If ๐‘Ž๐‘โˆˆโ„š and ๐‘› and ๐‘š are positive integers, then ๏€ป๐‘Ž๐‘๏‡ร—๏€ป๐‘Ž๐‘๏‡=๏€ป๐‘Ž๐‘๏‡.๏Š๏‰๏Š๏Šฐ๏‰
  • If ๐‘Ž๐‘โˆˆโ„š and ๐‘› and ๐‘š are positive integers with ๐‘›>๐‘š, then ๏€ป๐‘Ž๐‘๏‡รท๏€ป๐‘Ž๐‘๏‡=๏€ป๐‘Ž๐‘๏‡.๏Š๏‰๏Š๏Šฑ๏‰
  • If ๐‘Ž๐‘โˆˆโ„š and ๐‘› and ๐‘š are positive integers, then ๏€ผ๏€ป๐‘Ž๐‘๏‡๏ˆ=๏€ป๐‘Ž๐‘๏‡.๏Š๏‰๏Šร—๏‰
  • We can use the laws of exponents to simplify exponential expressions.

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