Skill: Apply the power-of-a-product rule

Explanation and Free Practice Resources

An exponent applied to an entire product applies to each factor: $(ab)^n=a^n b^n$ , for numerical or algebraic factors and whole-number exponents within this scope. The rule follows from interpreting powers as repeated multiplication and supports equivalent rewriting and simplification, while distinguishing products from sums: $(a+b)^n$ cannot generally be rewritten as $a^n+b^n$ . Fractional and negative exponents are outside this scope.

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Detailed Explanation: Apply the power-of-a-product rule

When a whole product is raised to a power, apply the exponent to each factor:

(ab)^n=a^n b^n

For example, simplify:

(3x)^4

Identify the two factors inside the parentheses: $3$ and $x$ .
Apply the exponent $4$ to both factors:

(3x)^4=3^4x^4

Calculate the numerical power:

3^4=81

Therefore,

\boxed{(3x)^4=81x^4}

This works because $(3x)^4$ means multiplying $3x$ by itself four times:

(3x)(3x)(3x)(3x)=3^4x^4

Remember that this rule applies to a product, not a sum. In general,

(a+b)^n\ne a^n+b^n

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Exponents - Power of a Product with Integer Base (Separate to Explicit Product)

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