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Rule

Power of a Product Property

A power with an integer exponent m whose base is the product of two non-zero factors a and b can be written as the product of two powers with bases a and b and the same exponent m.

(ab)^m = a^m b^m

For this rule to be valid when either a or b is 0, m must be greater than zero.

Proof

To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, (ab)^m will be written as a product using the definition of a power. (ab)^m=(ab)* (ab) * ⋯ * (ab)_(mtimes) Next, the parentheses can be removed. m times ↗ ↑ ↖ (ab)^m= a* b* a* b * ⋯ * a* b ↘ ↓ ↙ m times Now, the Commutative Property of Multiplication can be used. (ab)^m=a* a * ⋯ * a_(mtimes) * b* b * ⋯ * b_(mtimes) Finally, using the definition of a power again, the right-hand side of the equation can be written as the product of a^m and b^m.

(ab)^m = a^m b^m

The Power of a Product Property has been proven.

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