Powers of Products and Quotients – Practice Problems

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The powers of products rule says that raising a product a∙b to the power m, is the same as raising a to the power m and b to the power m and then considering the product a^m ∙ b^m:

(a ∙ b)^m = a^m ∙ b^m.

The powers of quotients rule says that raising a quotient a/b to the power m is the same as raising a to the power m and b to the power m and then considering the quotient a^m ÷ b^m:

(a ÷ b)^m = a^m ÷ b^m.

Indeed we can see in the following example that the powers of products rules works:
(3 ∙ 5)² = 3² ∙ 5²
(15)² = 9 ∙ 25
225 = 225

And in the following example that the powers of quotients rule also works:
(4 ÷ 2)³ = 4³ ÷ 2³
2³ = 64 ÷ 8
8 = 8

The powers of products and the powers of quotients rules are powerful tools which make solving algebraic and arithmetic problems easier and faster, allowing us to calculate important quantities in our daily lives, like speed or distance of moving objects.

Work with radicals and integer exponents.
CCSS.MATH.CONTENT.8.EE.A.1

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Exercises in this Practice Problem
Determine the missing term.
Prove that $(2\times 5)^3 = 2^5\times 3^5$ using the power of products rule.
Complete the following examples.
Determine which terms are equal.
Decide which example belongs to which rule.
Rewrite each term using the power of a product, power, and quotient rules.