Example

Simplifying (9d)2(-9d)^2 and (3mn)3(3mn)^3 Using the Product to a Power Property

Apply the Product to a Power Property to simplify two expressions in which a product of a number and one or more variables is raised to a power.

(9d)2=81d2(-9d)^2 = 81d^2: The base is the product 9d-9d, which has two factors: 9-9 and dd. Raise each factor to the second power using (ab)m=ambm(ab)^m = a^m b^m:

(9d)2=(9)2d2(-9d)^2 = (-9)^2 \cdot d^2

Simplify: (9)2=81(-9)^2 = 81 (an even exponent on a negative base produces a positive result), so the answer is 81d281d^2.

(3mn)3=27m3n3(3mn)^3 = 27m^3n^3: The base is the product 3mn3mn, which has three factors: 33, mm, and nn. Raise each factor to the third power:

(3mn)3=33m3n3(3mn)^3 = 3^3 \cdot m^3 \cdot n^3

Simplify: 33=273^3 = 27, so the answer is 27m3n327m^3n^3.

Both parts demonstrate the same procedure: identify every factor inside the parentheses, apply the exponent to each one individually, and then evaluate any numerical powers.

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Updated 2026-04-21

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