## Power of Products Property

## Introduction

An exponent is a mathematical way to show a repeated multiplication. The problem x to the sixth power can be shown as follows:

x^{6 }= x·x·x·x·x·x

At times, an exponent may apply to more than one variable. When this happens, the exponent can be applied to each variable individually.

(xyz)^{5} = xyz·xyz·xyz·xyz·xyz

= x·x·x·x·x · y·y·y·y·y · z·z·z·z·z

= x^{5}y^{5}z^{5}

## Lesson

Before discussing the main idea in this lesson, consider a similar property that you already know: the distributive property.

A mailman delivers mail to each mailbox along his route. Similarly, the distributive property distributes single number or term that is being multiplied by several terms. When an expression consists of several variables or numbers that are multiplied together, applying an exponent is done in much the same way as the distributive property.

When an exponent is applied to more than one number or variable, the exponent should be applied to each variable individually. The problem below contains the expanded work that shows why this is true.

Instead of doing all that work, there is a property that allows you to skip the repeated addition and use multiplication instead when taking several items in parenthesis to a single exponent. This property is called the power of products property.

The power of products property mentions two numbers or variables being multiplies, but it can also be used with products of 3, 4, 5, or more numbers. Some visual examples of how it works are given below:

Example #1 is a simple example of the power of products property.

**Example #1:** Simplify the expression (2xyz)^{4}

Using the property uses the same thinking as simply applying the rules of exponents. Simple problems such as the one above should only take a few seconds using the property. When tackling a problem with more variables and/or larger exponents, the value of remembering properties such as the power of products property is priceless.

**Example 2****:** Simplify the expression (3a^{4}b^{7}c^{12}d^{24})^{3}

The properties of exponents have names that are very similar. The ones you have learned thus far are:

When you are presented with a problem, it is easy to mix up the adding and multiplying involved in these properties. Some problems even use two or more of these properties. Try the following two examples yourself before revealing the answers.

Example 3: Simplify the expression 3x^{2}(y^{2}z^{3})^{4}·(xy)^{5}z^{2}

Example 4: Simplify the expression (2f^{3}g^{2}h)^{3}·5fg^{4}·(f^{2})^{3}(gh^{2})^{2}

These two examples use all three of the properties of exponents above. It helps to peek at these properties as you start out, then after successfully completing several problems you should try doing more problems without peeking at the properties. If you are working in a math book that has the answers in the back, then check the results as you do the problems. Knowing that you are doing the problems correctly will make you feel more comfortable with the material and confident in your abilities.

## Try It

Evaluate each expression:

1) (abc)^{4} =

2) 2(s^{2}t)^{2} =

3) (2s^{2}t)^{2} =

4) (d^{3}e^{4})^{3 }=

5) (3f^{3}g^{4})^{2} =

Evaluate each more complex expression:

6) 2r^{5}(s^{2}t^{5})^{2}·r^{4}(s^{2}t^{3})^{4 }=

7) (2x^{3}y^{2})^{2}·3y^{7}z^{3}·(x^{2})^{4}(z^{2})^{2} =

8) [(2m^{2}n^{3}p)^{2}]^{2}·[m^{2}(n^{3}p)^{3}]^{2} =

Scroll Down for Answers…

Answers:

1) (abc)4 = a^{4}b^{4}c^{4}

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

3) (2s^{2}t)^{2} = 4s^{4}t^{2}

4) (d^{3}e^{4})^{3 }= d^{9}e^{12}

5) (3f^{3}g^{4})^{2} = 9f^{6}g^{8}

Evaluate each more complex expression:

6) 2r^{5}(s^{2}t^{5})^{2}·r^{4}(s^{2}t^{3})^{4 }= 2r^{5}s^{4}t^{10 }· r^{4}s^{8}t^{12 }

^{ }= 2r^{9}s^{12}t^{22}

7) (2x^{3}y^{2})^{2}·3y^{7}z^{3}·(x^{2})^{4}(z^{2})^{2} = 4x^{6}y^{4 }· 3y^{7}z^{3 }· x^{8}z^{4}

= 12x^{14}y^{11}z^{7 }

8) [(2m^{2}n^{3}p)^{2}]^{2}·[m^{2}(n^{3}p)^{3}]^{2} = [4m^{4}n^{6}p^{2}]^{2 }· [m^{2}n^{9}p^{3}]^{2}

^{ }= 16m^{8}n^{12}p^{4 }· m^{4}n^{18}p^{6}

^{ }= 16m^{12}n^{30}p^{10}

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