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General Studies Math: Help & Review8 chapters | 85 lessons

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Lesson Transcript

Instructor:
*Jennifer Beddoe*

The Power of a Product rule states that a term raised to a power is equal to the product of its factors raised to the same power. In this lesson, learn more about this rule and look at some examples.

The **Power of a Product rule** is another way to simplify exponents. First, we need to define some terms as they relate to exponents. When you have a number or variable raised to a power, it is called the **base**, while the superscript number, or the number after the '^' mark, is called the **exponent** or **power**.

If there is more than one term in parenthesis, with an exponent outside the parenthesis, then the exponent is distributed to every term in the parenthesis. For example, consider this problem:

(*mn*)^5

Because of the parenthesis, the exponent (5) should be distributed to each term.

Therefore,

(*mn*)^5 = (*m*^5)(*n*^5)

There are two conditions that must be met in order for the Power of a Product rule to work.

- First, there must be two or more variables or constants that are being multiplied together. In the example we just looked at, those are the
*m*and*n*, but they could be any variable or constant. - Second, the result of the multiplication problem must be raised to a power. In the example, that is the 5.

Both conditions must be present in order to use the Power of a Product rule.

The Power of a Product rule can be proven by testing it using only numbers.

(4 * 2)^3

Using the Power of a Product rule, the solution is:

4^3 * 2^3 = 64 * 8 = 512

Then, work the problem like a simple math problem.

(4 * 2)^3 = 8^3 = 512

No matter what two numbers and exponent you use, the answer reached mathematically will always equal the answer found when you use the Power of a Product rule to solve it. Therefore, using the Power of a Product rule will also work when the problem contains variables.

You can use the Power of a Product rule for simple or more complex problems.

Simplify: (*ab*)^7

Since both conditions are met, there are two variables being multiplied together and the result of that multiplication is raised to a power, just distribute the exponent to both terms.

(*ab*)^7 = (*a*^7)(*b*^7)

Simplify: (2*xy*)^3

Distribute the 3 and simplify.

2^3 * *x*^3 * *y*^3 = (8*x*^3)(*y*^3)

If the terms in the parenthesis also have exponents, you can still simplify the problem. To distribute exponents, you multiply the exponents together.

(3*a*^2)^3

When you distribute the 3, you get:

3^3 * *a*^6

Since 2 * 3 = 6, the exponent with the *a* will be a 6.

The final answer is:

27*a*^6

Simplify: (4*m*^2*np*^3)^2

First, distribute the exponent that is outside the parenthesis.

(4^2)( *m*^2 * 2) (*n*^2) (*p*^3 * 2)

Then, simplify the exponents where possible. Remember that exponents raised to a power are multiplied together.

(16*m*^4)(*n*^2)(*p*^6)

Let's review. The **Power of a Product rule** is simply another way to simplify exponents. When simplifying exponents, if there is an expression with more than one term being multiplied together, and these terms are raised to a power, you can simplify the problem by distributing the exponent to every term in the parenthesis. If a term being multiplied already has an exponent, the two exponents are multiplied together.

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General Studies Math: Help & Review8 chapters | 85 lessons

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