## Power of Powers Property

## Introduction

A classic math problem involves putting 2 rabbits on a desert island, then imagining the number of rabbits doubling every month. As the months go by, the number of rabbits increases faster and faster. The number of rabbits can be represented by the equation r = 2^{m}, where r is the number of rabbits and m is the number of the month that they have been on the island.

This is a typical example of a problem that uses exponents. In this lesson you will learn not only about typical problems, but also about more complex problems. One such problem involves taking a power to another power. This type of problem looks like this:

You can use the same type of thinking to work through problems whether you are dealing with numbers or variables. This lesson explores how to evaluate any problem that contains an exponent taken to a higher exponent.

## Lesson

When learning rules of exponents, it is important to keep each of the rules straight. The previous lesson dealt with the product of powers, and here is a review of that rule:

As long as the bases are the same in the two terms being multiplied, you can simply add the exponents. Raising a power to another power is done in a slightly different way:

How about a shortcut? If you multiply the exponents in the first problem, 3 × 2 = 6. In the second problem, the exponents can also be multiplied: 7 × 4 = 28. The property that allows you to multiply the powers is called the power of powers property.

Example 1 demonstrates the power of powers property.

**Example 1****:** Use the power of powers property to evaluate each expression.

When dealing with numbers, there may be more than one way to do the problem. Example 2 shows two different ways to evaluate the problem.

**Example 2****:** Evaluate the expression (2^{2})^{2} in two different ways.

The problems so far have been pretty basic. Here is an example that is a little more involved.

**Example 3****:** Evaluate the expression.

## Try It

Evaluate each expression:

1) (r^{2})^{3} =

2) (s^{4})^{5} =

3) (t^{10})^{6 }=

4) (u^{2})^{5 }∙ u^{6 }=

5) (v^{4})^{2}z^{7 }∙ (v^{3})^{5}z^{4 }=

6) a^{3}b^{6}c^{8} ∙ (a^{4})^{3}b^{2}(c^{5})^{3} =

7) [(w^{2})^{2}]^{2}

Scroll Down for Answers…

Answers:

1) (r^{2})^{3} = r^{6}

2) (s^{4})^{5} = s^{20}

3) (t^{10})^{6 }= t^{60}

4) (u^{2})^{5 }∙ u^{6 }= u^{16}

5) (v^{4})^{2}z^{7 }∙ (v^{3})^{5}z^{4 }= v^{23}z^{11}

6) a^{3}b^{6}c^{8} ∙ (a^{4})^{3}b^{2}(c^{5})^{5} = a^{3}b^{6}c^{8} ∙ a^{12}b^{2}c^{25} = a^{15}b^{8}c^{33}

7) [(w^{2})^{2}]^{2} = [w^{4}]^{2} = w^{8}

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