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

Explanation and Free Practice Resources

The power-of-a-power relationship $(a^m)^n=a^{mn}$ expresses that raising an existing power to another whole-number power repeats the entire group of $m$ factors $n$ times, so the exponents are multiplied rather than added. This applies to numerical and algebraic bases with whole-number exponents and supports simplifying exponential expressions; negative, fractional, or irrational exponents and their associated edge cases are beyond this scope.

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

When a power is raised to another power, multiply the exponents:

(a^m)^n=a^{m\cdot n}

The outside exponent means the entire power is repeated that many times.

Example

Simplify:

(x^3)^4

Step 1: Identify the exponents.

The inside exponent is $3$ .
The outside exponent is $4$ .

Step 2: Multiply the exponents.

3\cdot 4=12

Step 3: Write the simplified power.

(x^3)^4=x^{12}

So, the answer is:

\boxed{x^{12}}

Do not add the exponents. The outside power repeats all $3$ factors four times, giving $3\cdot4=12$ factors.

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Exponents - Power Law with Variable Base (Positives, Exponent with Power to Exponent)

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