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Lesson Explainer: Laws of Exponents over the Real Numbers Mathematics • 9th Grade

In this explainer, we will learn how to apply the laws of exponents to multiply and divide powers and to calculate a power raised to a power over the real numbers.

Recall that to evaluate a power, we multiply the base by the exponent the number of times indicated. Letโ€™s recap the definition below.

Definition: Evaluating a Power

For a power with base ๐‘Ž, ๐‘Žโˆˆโ„, and exponent ๐‘›, ๐‘›โˆˆโ„ค๏Šฐ, then ๐‘Ž=๐‘Žร—๐‘Žร—๐‘Žร—โ‹ฏร—๐‘Ž.๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Ž๏Š๏Štimes

By using our understanding of how to evaluate power, we can derive the rules of how to multiply and divide powers as well as the laws for a power raised to a power.

Letโ€™s say we have two powers with base ๐‘Ž, ๐‘Žโˆˆโ„, where the first is raised to the power of ๐‘š and the second is raised to the power of ๐‘›, ๐‘š, ๐‘›โˆˆโ„ค. If we multiply the two powers together, we get ๐‘Žร—๐‘Ž=(๐‘Žร—๐‘Žร—๐‘Žร—โ‹ฏร—๐‘Ž)๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Žร—(๐‘Žร—๐‘Žร—๐‘Žร—โ‹ฏร—๐‘Ž)๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Ž.๏‰๏Š๏‰๏Štimestimes

If we count the number of times ๐‘Ž is multiplied by itself, this is ๐‘š times plus ๐‘› times, which is a total of ๐‘š+๐‘› times. This then gives us ๐‘Žร—๐‘Ž=๐‘Ž,๏‰๏Š๏‰๏Šฐ๏Š which is the product rule for exponents.

Now, letโ€™s say we have two powers with base ๐‘Ž, ๐‘Žโˆˆโ„โˆ’0, where the first is raised to the power of ๐‘š and the second is raised to the power of ๐‘›, ๐‘šโˆˆโ„ค, ๐‘›โˆˆโ„ค. If we divide the first power by the second power, we get ๐‘Žรท๐‘Ž=(๐‘Žร—๐‘Žร—๐‘Žร—โ‹ฏร—๐‘Ž)รท(๐‘Žร—๐‘Žร—๐‘Žร—โ‹ฏร—๐‘Ž)=๏‡„๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏‡…๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏‡†๐‘Žร—๐‘Žร—๐‘Žร—๐‘Žร—๐‘Žร—โ‹ฏร—๐‘Ž๐‘Žร—๐‘Žร—๐‘Žร—โ‹ฏร—๐‘Ž๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Ž.๏‰๏Š๏‰๏Štimestimes

If we cancel common factors in the numerator and the denominator, we end up canceling all the ๐‘Žโ€™s in the numerator, meaning we have ๐‘š lots of ๐‘Ž minus ๐‘› lots of ๐‘Ž, leaving us with ๐‘šโˆ’๐‘› lots of ๐‘Ž. In other words, we have =๏‡„๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏‡…๏†ช๏†ช๏†ช๏†ช๏†ช๏†ช๏‡†๐‘Žร—๐‘Žร—๐‘Žร—๐‘Žร—๐‘Žโ‹ฏร—๐‘Ž๐‘Žร—๐‘Žร—๐‘Žร—โ‹ฏร—๐‘Ž๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Ž=๐‘Žร—๐‘Žร—โ‹ฏร—๐‘Ž๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏‡Ž.๏‰๏Š๏‰๏Šฑ๏Štimestimestimes

This then gives us, ๐‘Žรท๐‘Ž=๐‘Ž,๏‰๏Š๏‰๏Šฑ๏Š when ๐‘šโ‰ฅ๐‘›.

If we consider the case when ๐‘› is larger than ๐‘š, so ๐‘š<๐‘›, then we can use the same approach, but with a higher power in the denominator, giving us ๐‘Žรท๐‘Ž=(๐‘Žร—๐‘Žร—๐‘Žโ‹ฏร—๐‘Ž)รท(๐‘Žร—๐‘Žร—๐‘Žร—๐‘Žร—๐‘Žโ‹ฏร—๐‘Ž)=๏‡„๏†ช๏†ช๏†ช๏†ช๏‡…๏†ช๏†ช๏†ช๏†ช๏‡†๐‘Žร—๐‘Žร—๐‘Žร—โ‹ฏร—๐‘Ž๐‘Žร—๐‘Žร—๐‘Žร—๐‘Žร—๐‘Žโ‹ฏร—๐‘Ž๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Ž.๏‰๏Š๏‰๏Štimestimes

Canceling common factors, we get =๏‡„๏†ช๏†ช๏†ช๏†ช๏‡…๏†ช๏†ช๏†ช๏†ช๏‡†๐‘Žร—๐‘Žร—๐‘Žร—โ‹ฏร—๐‘Ž๐‘Žร—๐‘Žร—๐‘Žร—๐‘Žร—๐‘Žโ‹ฏร—๐‘Ž๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Ž=1๐‘Žร—๐‘Žร—โ‹ฏร—๐‘Ž๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏‡Ž=1๐‘Ž.๏‰๏Š๏Š๏Šฑ๏‰๏Š๏Šฑ๏‰timestimestimes

If we then use the law for negative indices, we get =๐‘Ž.๏‰๏Šฑ๏Š

Therefore, ๐‘Žรท๐‘Ž=๐‘Ž๏‰๏Š๏‰๏Šฑ๏Š for all integer values of ๐‘š and ๐‘›, which is the quotient rule for exponents.

Next, letโ€™s say we have a power with a base of ๐‘Ž๐‘, ๐‘Ž, ๐‘โˆˆโ„, which is raised to the power of ๐‘›, ๐‘›โˆˆโ„ค. This gives us (๐‘Ž๐‘)=๐‘Ž๐‘ร—๐‘Ž๐‘ร—๐‘Ž๐‘โ‹ฏร—๐‘Ž๐‘๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Ž.๏Š๏Štimes

If we rewrite this so that all of the ๐‘Žโ€™s and all of the ๐‘โ€™s are together, we get (๐‘Ž๐‘)=(๐‘Žร—๐‘Žร—๐‘Žโ‹ฏร—๐‘Ž)๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Žร—(๐‘ร—๐‘ร—๐‘โ‹ฏร—๐‘)๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Ž.๏Š๏Š๏Štimestimes

Since the number of times ๐‘Ž is multiplied by itself is ๐‘› times and similarly the number of times ๐‘ is multiplied by itself is ๐‘› times, then we get (๐‘Ž๐‘)=๐‘Ž๐‘,๏Š๏Š๏Š which is the power of a product rule for exponents.

Similarly, letโ€™s say we have a power with a base of ๐‘Ž๐‘, ๐‘Žโˆˆโ„, ๐‘โˆˆโ„โˆ’0, which is raised to the power of ๐‘›, ๐‘›โˆˆโ„ค. This gives us ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘ร—๐‘Ž๐‘ร—๐‘Ž๐‘โ‹ฏร—๐‘Ž๐‘๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Ž.๏Š๏Štimes

If we rewrite this as one fraction, with the ๐‘Žโ€™s in the numerator and the ๐‘โ€™s in the denominator, we get ๏€ป๐‘Ž๐‘๏‡=๏‡„๏†ช๏†ช๏†ช๏†ช๏‡…๏†ช๏†ช๏†ช๏†ช๏‡†๐‘Žร—๐‘Žร—๐‘Žโ‹ฏร—๐‘Ž๐‘ร—๐‘ร—๐‘โ‹ฏร—๐‘๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Ž.๏Š๏Š๏Štimestimes

Since the number of times ๐‘Ž is multiplied by itself is ๐‘› times and similarly the number of times ๐‘ is multiplied by itself is ๐‘› times, then we get ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘,๏Š๏Š๏Š which is the power of a quotient rule for exponents.

Lastly, if we have now only one power with base ๐‘Ž, ๐‘Žโˆˆโ„, which is raised to the power of ๐‘š and then raised to another power ๐‘›, ๐‘š, ๐‘›โˆˆโ„ค, we get the following: (๐‘Ž)=(๐‘Ž)ร—(๐‘Ž)ร—(๐‘Ž)โ‹ฏร—(๐‘Ž)๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Ž=(๐‘Žร—๐‘Žร—๐‘Žโ‹ฏร—๐‘Ž)๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Žร—(๐‘Žร—๐‘Žร—๐‘Žโ‹ฏร—๐‘Ž)๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Žร—(๐‘Žร—๐‘Žร—๐‘Žโ‹ฏร—๐‘Ž)๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Žโ‹ฏร—(๐‘Žร—๐‘Žร—๐‘Žโ‹ฏร—๐‘Ž)๏‡Œ๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡๏†ฒ๏†ฒ๏†ฒ๏†ฒ๏‡Ž.๐‘›๐‘š๏‰๏Š๏‰๏‰๏‰๏‰๏Š๏‰๏‰๏‰๏‰timestimestimestimestimeslotsoftimes

If we count the number of times ๐‘Ž is multiplied by itself, this is ๐‘š times multiplied by ๐‘› times, which is a total of ๐‘š๐‘› times. This then gives us (๐‘Ž)=๐‘Ž,๏‰๏Š๏‰๏Š which is called the power rule for exponents.

Letโ€™s summarize the laws in the rules below.

Rule: Laws of Exponents

The following are the rules of exponents with real bases and integer exponents:

  • The product rule: ๐‘Žร—๐‘Ž=๐‘Ž,๏‰๏Š๏‰๏Šฐ๏Š where ๐‘Žโˆˆโ„ and ๐‘š,๐‘›โˆˆโ„ค.
  • The quotient rule: ๐‘Žรท๐‘Ž=๐‘Ž,๏‰๏Š๏‰๏Šฑ๏Š where ๐‘Žโˆˆโ„โˆ’0, ๐‘šโˆˆโ„ค, and ๐‘›โˆˆโ„ค.
  • The power of a product rule: (๐‘Ž๐‘)=๐‘Ž๐‘,๏Š๏Š๏Š where ๐‘Ž, ๐‘โˆˆโ„ and ๐‘›โˆˆโ„ค.
  • The power of a quotient rule: ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘,๏Š๏Š๏Š where ๐‘Žโˆˆโ„, ๐‘โˆˆโ„โˆ’0, and ๐‘›โˆˆโ„ค.
  • The power rule: (๐‘Ž)=๐‘Ž,๏‰๏Š๏‰๏Š where ๐‘Žโˆˆโ„ and ๐‘š, ๐‘›โˆˆโ„ค.

As we are working with bases with real numbers, then often we need to simplify expressions with square roots. As such, it is helpful to recall the following definition.

Definition: Simplifying Expressions with Squares and Square Roots

For an expression with a base ๐‘Ž, where ๐‘Ž>0,

  • ๏€บโˆš๐‘Ž๏†=๐‘Ž๏Šจ;
  • โˆš๐‘Ž=๐‘Ž๏Šจ.

In our first example, we will discuss how to use the product rule for exponents to simplify an expression with a real base.

Example 1: Simplifying an Expression Involving Multiplication of Positive Integer Powers in the Real Numbers

Fill in the blank: The expression โˆš3ร—๏€ปโˆš3๏‡ร—๏€ปโˆš3๏‡๏Šจ๏Šฉ simplifies to .

Answer

To find what โˆš3ร—๏€ปโˆš3๏‡ร—๏€ปโˆš3๏‡๏Šจ๏Šฉ is equal to, we need to simplify the expression. As we have powers with the same base being multiplied together, then we can use the product rule for exponents, which states that ๐‘Žร—๐‘Ž=๐‘Ž,๏‰๏Š๏‰๏Šฐ๏Š where ๐‘Žโˆˆโ„ and ๐‘š, ๐‘›โˆˆโ„ค.

Starting with โˆš3ร—๏€ปโˆš3๏‡๏Šจ, we can write this as ๏€ปโˆš3๏‡ร—๏€ปโˆš3๏‡๏Šง๏Šจ. Then, applying the product rule, we get ๏€ปโˆš3๏‡ร—๏€ปโˆš3๏‡=๏€ปโˆš3๏‡=๏€ปโˆš3๏‡.๏Šง๏Šจ๏Šง๏Šฐ๏Šจ๏Šฉ

Substituting ๏€ปโˆš3๏‡๏Šฉ back into the original expression, we get โˆš3ร—๏€ปโˆš3๏‡ร—๏€ปโˆš3๏‡=๏€ปโˆš3๏‡ร—๏€ปโˆš3๏‡.๏Šจ๏Šฉ๏Šฉ๏Šฉ

Again, we can apply the product rule, which gives us ๏€ปโˆš3๏‡ร—๏€ปโˆš3๏‡=๏€ปโˆš3๏‡=๏€ปโˆš3๏‡.๏Šฉ๏Šฉ๏Šฉ๏Šฐ๏Šฉ๏Šฌ

Lastly, to simplify ๏€ปโˆš3๏‡๏Šฌ, we can use the property ๏€บโˆš๐‘Ž๏†=๐‘Ž๏Šจ, where ๐‘Ž>0. To do so, we can write ๏€ปโˆš3๏‡๏Šฌ as a product of powers of 2 using the product law in reverse. This gives us ๏€ปโˆš3๏‡=๏€ปโˆš3๏‡ร—๏€ปโˆš3๏‡ร—๏€ปโˆš3๏‡=3ร—3ร—3=27.๏Šฌ๏Šจ๏Šจ๏Šจ

Therefore, โˆš3ร—๏€ปโˆš3๏‡ร—๏€ปโˆš3๏‡๏Šจ๏Šฉ simplifies to 27.

In the next example, we will use the quotient rule for exponents to simplify an expression with a real base.

Example 2: Simplifying an Expression Involving Division of Positive Integer Powers in the Real Numbers

Which of the following is equal to ๏€ปโˆš7๏‡รท๏€ปโˆš7๏‡๏Šฎ๏Šฌ?

  1. 7๏Žฃ๏Žข
  2. 7๏Šจ
  3. ๏€ปโˆš7๏‡๏Šง๏Šช
  4. ๏€ปโˆš7๏‡๏Šจ
  5. ๏€ปโˆš7๏‡๏Žฃ๏Žข

Answer

To find what ๏€ปโˆš7๏‡รท๏€ปโˆš7๏‡๏Šฎ๏Šฌ is equal to, we need to simplify the expression. As we have two powers with the same base, where one is divided by the other, then we can use the quotient rule for exponents, which states that ๐‘Žรท๐‘Ž=๐‘Ž,๏‰๏Š๏‰๏Šฑ๏Š where ๐‘Žโˆˆโ„โˆ’0, ๐‘šโˆˆโ„ค, and ๐‘›โˆˆโ„ค.

Applying this rule with ๐‘Ž=โˆš7, ๐‘š=8, and ๐‘›=6, we get ๏€ปโˆš7๏‡รท๏€ปโˆš7๏‡=๏€ปโˆš7๏‡=๏€ปโˆš7๏‡.๏Šฎ๏Šฌ๏Šฎ๏Šฑ๏Šฌ๏Šจ

Comparing with the options given, we can see that this is the same as option D, which is the correct answer.

In the next example, we will consider how to use rules for exponents to simplify expressions with square roots in them, where one base is the negative of another.

Example 3: Simplifying an Expression Involving Division of Positive Integer Powers in the Real Numbers

What is the value of ๏€ปโˆ’โˆš3๏‡รท๏€ปโˆš3๏‡๏Šฌ๏Šฉ?

Answer

To find the value of ๏€ปโˆ’โˆš3๏‡รท๏€ปโˆš3๏‡๏Šฌ๏Šฉ, we need to first simplify the expression. One approach is to use the laws of exponents.

We can see that the bases are nearly the same for the two parts of the expression, except that the first base is the negative of the second base. Instead, we can write the expression as ๏€ปโˆ’โˆš3๏‡รท๏€ปโˆš3๏‡=๏€ปโˆ’1ร—โˆš3๏‡รท๏€ปโˆš3๏‡.๏Šฌ๏Šฉ๏Šฌ๏Šฉ

Using the power of a product rule for exponents, which states that (๐‘Ž๐‘)=๐‘Ž๐‘,๏Š๏Š๏Š where ๐‘Ž, ๐‘โˆˆโ„ and ๐‘›โˆˆโ„ค, we can apply this to ๏€ปโˆ’1ร—โˆš3๏‡๏Šฌ, giving us ๏€ปโˆ’1ร—โˆš3๏‡รท๏€ปโˆš3๏‡=(โˆ’1)๏€ปโˆš3๏‡รท๏€ปโˆš3๏‡.๏Šฌ๏Šฉ๏Šฌ๏Šฌ๏Šฉ

Since ๏€ปโˆš3๏‡๏Šฌ and ๏€ปโˆš3๏‡๏Šฉ now have the same base, we can apply the quotient rule for exponents, which states that ๐‘Žรท๐‘Ž=๐‘Ž,๏‰๏Š๏‰๏Šฑ๏Š where ๐‘Žโˆˆโ„โˆ’0 and ๐‘š, ๐‘›โˆˆโ„ค, giving us (โˆ’1)๏€ปโˆš3๏‡รท๏€ปโˆš3๏‡=(โˆ’1)๏€ปโˆš3๏‡=(โˆ’1)๏€ปโˆš3๏‡.๏Šฌ๏Šฌ๏Šฉ๏Šฌ๏Šฌ๏Šฑ๏Šฉ๏Šฌ๏Šฉ

We can now evaluate this, by calculating (โˆ’1)๏Šฌ and using ๏€บโˆš๐‘Ž๏†=๐‘Ž๏Šจ and the product rule for exponents to find ๏€ปโˆš3๏‡๏Šฉ, as follows: (โˆ’1)๏€ปโˆš3๏‡=1ร—๏€ปโˆš3๏‡=๏€ปโˆš3๏‡ร—โˆš3=3โˆš3.๏Šฌ๏Šฉ๏Šฉ๏Šจ

Therefore, the value of ๏€ปโˆ’โˆš3๏‡รท๏€ปโˆš3๏‡๏Šฌ๏Šฉ is 3โˆš3.

As with mathematical operations in general, we can apply multiple operations in one expression and use the laws of exponents to simplify these. We usually apply the laws according to the order of operations. We will explore this further in the next example.

Example 4: Evaluating a Numerical Expression Involving Real Numbers Using the Law of Exponents

Simplify ๏€ปโˆš5๏‡ร—๏€ปโˆš5๏‡๏€ปโˆš5๏‡๏Šฌ๏Šจ๏Šช.

Answer

To simplify ๏€ปโˆš5๏‡ร—๏€ปโˆš5๏‡๏€ปโˆš5๏‡๏Šฌ๏Šจ๏Šช, we need to use the laws of exponents. To do so, it is helpful to apply the laws to the numerator first, which is ๏€ปโˆš5๏‡ร—๏€ปโˆš5๏‡๏Šฌ๏Šจ.

To simplify ๏€ปโˆš5๏‡ร—๏€ปโˆš5๏‡๏Šฌ๏Šจ, we can use the product rule for exponents, which states that ๐‘Žร—๐‘Ž=๐‘Ž,๏‰๏Š๏‰๏Šฐ๏Š where ๐‘Žโˆˆโ„, ๐‘š, ๐‘›โˆˆโ„ค, giving us ๏€ปโˆš5๏‡ร—๏€ปโˆš5๏‡=๏€ปโˆš5๏‡=๏€ปโˆš5๏‡.๏Šฌ๏Šจ๏Šฌ๏Šฐ๏Šจ๏Šฎ

Substituting this back into the expression, we get ๏€ปโˆš5๏‡๏€ปโˆš5๏‡๏Šฎ๏Šช. To simplify further, we can use the quotient rule for exponents, which states that ๐‘Žรท๐‘Ž=๐‘Ž,๏‰๏Š๏‰๏Šฑ๏Š where ๐‘Žโˆˆโ„โˆ’0, ๐‘šโˆˆโ„ค, and ๐‘›โˆˆโ„ค, giving us ๏€ปโˆš5๏‡๏€ปโˆš5๏‡=๏€ปโˆš5๏‡=๏€ปโˆš5๏‡.๏Šฎ๏Šช๏Šฎ๏Šฑ๏Šช๏Šช

We can use the rule ๏€บโˆš๐‘Ž๏†=๐‘Ž๏Šจ and the product rule to simplify this further, giving us ๏€ปโˆš5๏‡=๏€ปโˆš5๏‡ร—๏€ปโˆš5๏‡=5ร—5=25.๏Šช๏Šจ๏Šจ

Therefore, ๏€ปโˆš5๏‡ร—๏€ปโˆš5๏‡๏€ปโˆš5๏‡๏Šฌ๏Šจ๏Šช simplified gives us 25.

Next, we will consider how to simplify expressions where there are negative exponents. Letโ€™s recall the laws of zero and negative exponents.

Laws: Rules for Zero and Negative Exponents

  • The law for zero exponents: ๐‘Ž=1,๏Šฆ where ๐‘Žโˆˆโ„โˆ’0.
  • The law for negative exponents: ๐‘Ž=1๐‘Ž๐‘Ž=1๐‘Ž,๏Šฑ๏Š๏Š๏Š๏Šฑ๏Šor where ๐‘Žโˆˆโ„โˆ’0 and ๐‘›โˆˆโ„ค.

Using the law for negative exponents, we can derive two other properties.

First, if we take ๐‘Ž=1๐‘Ž๏Šฑ๏Š๏Š, where ๐‘Žโˆˆโ„โˆ’0 and ๐‘›โˆˆโ„ค, and multiply both sides by ๐‘Ž๏Š, we get ๐‘Ž=1๐‘Ž๐‘Žร—๐‘Ž=1.๏Šฑ๏Š๏Š๏Š๏Šฑ๏Š

Second, if we apply the law for negative exponents to a power with base ๐‘Ž๐‘, where ๐‘Ž, ๐‘โˆˆโ„โˆ’0 and exponent ๐‘›โˆˆโ„ค, we get ๏€ป๐‘Ž๐‘๏‡=1๏€ป๏‡.๏Š๏Œบ๏Œป๏Šฑ๏Š

By the power of a quotient rule, which states that ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘,๏Š๏Š๏Š where ๐‘Žโˆˆโ„, ๐‘โˆˆโ„โˆ’0, and ๐‘›โˆˆโ„ค, we then get ๏€ป๐‘Ž๐‘๏‡=1.๏Š๏Œบ๏Œป๏Žช๏‘ƒ๏Žช๏‘ƒ

If we multiply the numerator and denominator by ๐‘๏Šฑ๏Š, we then get ๏€ป๐‘Ž๐‘๏‡=1ร—๐‘๐‘=๐‘ร—๐‘=๐‘๐‘Ž.๏Š๏Œบ๏Œป๏Šฑ๏Š๏Šฑ๏Š๏Šฑ๏Š๏Œบ๏Œป๏Šฑ๏Š๏Šฑ๏Š๏Šฑ๏Š๏Žช๏‘ƒ๏Žช๏‘ƒ๏Žช๏‘ƒ๏Žช๏‘ƒ

If we then apply the power of a quotient rule again, we get ๏€ป๐‘Ž๐‘๏‡=๏€ฝ๐‘๐‘Ž๏‰.๏Š๏Šฑ๏Š

Similarly, we can show that ๏€ป๐‘Ž๐‘๏‡=๏€ฝ๐‘๐‘Ž๏‰.๏Šฑ๏Š๏Š

Letโ€™s summarize these two rules.

Laws: Rules for Zero and Negative Exponents

  • The law of multiplicative inverses: ๐‘Žร—๐‘Ž=1,๏Š๏Šฑ๏Š where ๐‘Žโˆˆโ„โˆ’0 and ๐‘›โˆˆโ„ค.
  • The power of a quotient rule for negative exponents: ๏€ป๐‘Ž๐‘๏‡=๏€ฝ๐‘๐‘Ž๏‰๏€ป๐‘Ž๐‘๏‡=๏€ฝ๐‘๐‘Ž๏‰,๏Š๏Šฑ๏Š๏Šฑ๏Š๏Šor where ๐‘Ž, ๐‘โˆˆโ„โˆ’0 and ๐‘›โˆˆโ„ค.

In the next example, we will use the law for negative exponents and the laws of indices to evaluate an expression.

Example 5: Evaluating a Numerical Expression Involving Real Numbers Using the Law of Exponents

For ๐‘Ž=154 and ๐‘=52, evaluate ๐‘Ž๐‘๏Šจ๏Šฑ๏Šฉ.

Answer

To evaluate ๐‘Ž๐‘๏Šจ๏Šฑ๏Šฉ, we must start by substituting ๐‘Ž=154 and ๐‘=52. This gives us ๐‘Ž๐‘=๏€ผ154๏ˆ๏€ผ52๏ˆ.๏Šจ๏Šฑ๏Šฉ๏Šจ๏Šฑ๏Šฉ

Next, we can use the power of a quotient rule for negative exponents to simplify ๏€ผ52๏ˆ๏Šฑ๏Šฉ, which states that ๏€ป๐‘Ž๐‘๏‡=๏€ฝ๐‘๐‘Ž๏‰๏€ป๐‘Ž๐‘๏‡=๏€ฝ๐‘๐‘Ž๏‰,๏Š๏Šฑ๏Š๏Šฑ๏Š๏Šor where ๐‘Ž, ๐‘โˆˆโ„โˆ’0 and ๐‘›โˆˆโ„ค, giving us ๏€ผ52๏ˆ=๏€ผ25๏ˆ.๏Šฑ๏Šฉ๏Šฉ

Substituting this back to the original expression, we have ๐‘Ž๐‘=๏€ผ154๏ˆร—๏€ผ25๏ˆ.๏Šจ๏Šฑ๏Šฉ๏Šจ๏Šฉ

Using the power of a quotient rule, which states that ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘,๏Š๏Š๏Š where ๐‘Žโˆˆโ„, ๐‘โˆˆโ„โˆ’0, and ๐‘›โˆˆโ„ค, we get ๏€ผ154๏ˆ=154๏Šจ๏Šจ๏Šจ and ๏€ผ25๏ˆ=25.๏Šฉ๏Šฉ๏Šฉ

This then gives us ๐‘Ž๐‘=154ร—25.๏Šจ๏Šฑ๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šฉ

If we expand the powers, we get ๐‘Ž๐‘=154ร—154ร—25ร—25ร—25.๏Šจ๏Šฑ๏Šฉ

Canceling common factors then gives us ๐‘Ž๐‘=154ร—154ร—25ร—25ร—25๐‘Ž๐‘=3ร—32ร—5=910.๏Šจ๏Šฑ๏Šฉ๏Šฉ๏Šฉ๏Šจ๏Šจ๏Šฑ๏Šฉ

Therefore, for ๐‘Ž=154 and ๐‘=52, the value of ๐‘Ž๐‘๏Šจ๏Šฑ๏Šฉ is 910.

So far, we have considered how to use the laws of exponents to simplify expressions with real bases. Next, we will consider how to solve simple exponential equations using the laws of exponents. Recall that if two powers are equal, then if either the bases or the exponents are also equal, the following rules apply.

Rule: Equating Powers with the Same Base or the Same Exponent

  • If ๐‘Ž=๐‘Ž๏‰๏Š, where ๐‘Žโˆˆโ„โˆ’[โˆ’1,0,1], then ๐‘š=๐‘›.
  • If ๐‘Ž=๐‘๏Š๏Š, then ๐‘Ž=๐‘ when ๐‘›โˆˆ[1,3,5,โ€ฆ] and |๐‘Ž|=|๐‘| when ๐‘›โˆˆ[2,4,6,โ€ฆ].

In the next example, we will discuss how to use the laws of exponents to solve simple exponential equations.

Example 6: Using the Laws of Exponents to Solve a Simple Exponential Equation

Find the value of ๐‘ฅ in the equation 16ร—2=๏€น2๏…๏Šฏ๏Šฑ๏—๏—๏Šฑ๏Šง๏Šจ.

Answer

To find the value of ๐‘ฅ in the equation 16ร—2=๏€น2๏…๏Šฏ๏Šฑ๏—๏—๏Šฑ๏Šง๏Šจ, we can use the laws of exponents to simplify the different parts of the equation and then use the rule that states that if ๐‘Ž=๐‘Ž๏‰๏Š, where ๐‘Žโˆˆโ„โˆ’[โˆ’1,0,1], then ๐‘š=๐‘›, to find the value of ๐‘ฅ.

In order to apply the rules of equal powers, we need to have two powers with either the same base or the same exponent. As here the unknown is in the exponents, we need to write each side of the equation as a power with the same base. Meaning, we need to write 16 as 2๏Šช, giving us 2ร—2,๏Šช๏Šฏ๏Šฑ๏— which we can simplify using the product rule, which states that ๐‘Žร—๐‘Ž=๐‘Ž,๏‰๏Š๏‰๏Šฐ๏Š where ๐‘Žโˆˆโ„ and ๐‘š, ๐‘›โˆˆโ„ค.

This then gives us on the left-hand side 2ร—2=2=2.๏Šช๏Šฏ๏Šฑ๏—๏Šช๏Šฐ๏Šฏ๏Šฑ๏—๏Šง๏Šฉ๏Šฑ๏—

Looking at the right-hand side, ๏€น2๏…๏—๏Šฑ๏Šง๏Šจ, we can simplify this using the power rule, which states that (๐‘Ž)=๐‘Ž,๏‰๏Š๏‰๏Š where ๐‘Žโˆˆโ„ and ๐‘š, ๐‘›โˆˆโ„ค.

This then gives us on the right-hand side ๏€น2๏…=2=2.๏—๏Šฑ๏Šง๏Šจ(๏—๏Šฑ๏Šง)ร—๏Šจ๏Šจ๏—๏Šฑ๏Šจ

Equating both sides, we get 2=2.๏Šง๏Šฉ๏Šฑ๏—๏Šจ๏—๏Šฑ๏Šจ

Since the bases are both equal, then the exponents must also be equal, giving us 13โˆ’๐‘ฅ=2๐‘ฅโˆ’2.

Solving for ๐‘ฅ, we then get 13โˆ’๐‘ฅ=2๐‘ฅโˆ’215โˆ’๐‘ฅ=2๐‘ฅ3๐‘ฅ=15๐‘ฅ=5.

Therefore, the value of ๐‘ฅ is 5 in the equation 16ร—2=๏€น2๏…๏Šฏ๏Šฑ๏—๏—๏Šฑ๏Šง๏Šจ.

In this explainer, we have learned how to use the laws of exponents to simplify expressions with real numbers raised to integer powers. We have also learned how to solve simple exponential equations using the laws of powers. Letโ€™s recap the key points.

Key Points

  • We can use the laws of exponents to simplify powers with real bases and integer powers. These rules state the following:
    • The product rule: ๐‘Žร—๐‘Ž=๐‘Ž,๏‰๏Š๏‰๏Šฐ๏Š where ๐‘Žโˆˆโ„ and ๐‘š, ๐‘›โˆˆโ„ค.
    • The quotient rule: ๐‘Žรท๐‘Ž=๐‘Ž,๏‰๏Š๏‰๏Šฑ๏Š where ๐‘Žโˆˆโ„โˆ’0, ๐‘šโˆˆโ„ค, and ๐‘›โˆˆโ„ค.
    • The power of a product rule: (๐‘Ž๐‘)=๐‘Ž๐‘,๏Š๏Š๏Š where ๐‘Ž, ๐‘โˆˆโ„ and ๐‘›โˆˆโ„ค.
    • The power of a quotient rule: ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘,๏Š๏Š๏Š where ๐‘Žโˆˆโ„, ๐‘โˆˆโ„โˆ’0, and ๐‘›โˆˆโ„ค.
    • The power rule: (๐‘Ž)=๐‘Ž,๏‰๏Š๏‰๏Š where ๐‘Žโˆˆโ„ and ๐‘š, ๐‘›โˆˆโ„ค.
    • The law of multiplicative inverses: ๐‘Žร—๐‘Ž=1,๏Š๏Šฑ๏Š where ๐‘Žโˆˆโ„โˆ’0 and ๐‘›โˆˆโ„ค.
    • The power of a quotient rule for negative exponents: ๏€ป๐‘Ž๐‘๏‡=๏€ฝ๐‘๐‘Ž๏‰๏€ป๐‘Ž๐‘๏‡=๏€ฝ๐‘๐‘Ž๏‰,๏Š๏Šฑ๏Š๏Šฑ๏Š๏Šor where ๐‘Ž, ๐‘โˆˆโ„โˆ’0 and ๐‘›โˆˆโ„ค.
  • To solve simple exponential equations with real bases, we can use the following rules when the bases or exponents are equal:
    • If ๐‘Ž=๐‘Ž๏‰๏Š, where ๐‘Žโˆˆโ„โˆ’[โˆ’1,0,1], then ๐‘š=๐‘›.
    • If ๐‘Ž=๐‘๏Š๏Š, then ๐‘Ž=๐‘ when ๐‘›โˆˆ[1,3,5,โ€ฆ] and |๐‘Ž|=|๐‘| when ๐‘›โˆˆ[2,4,6,โ€ฆ].

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