Apply the power-of-a-power rule

The power-of-a-power relationship $(a^m)^n=a^{mn}$ expresses that raising a power to another whole-number power repeats the original multiplication, so the exponents are multiplied rather than added. It applies to numerical and algebraic bases, including powers of 10 in scientific notation, with parentheses preserving the base; rational or irrational exponents and more advanced exponent definitions are outside this scope.

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

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

(a^m)^n=a^{mn}

The parentheses show that the entire power is being raised again. Do not add the exponents.

Example:

(5^2)^3

Identify the exponents: $m=2$ and $n=3$ .
Multiply them:

2\cdot 3=6

Write the result with the same base, $5$ :

(5^2)^3=5^6

So,

\boxed{(5^2)^3=5^6}

This works because $(5^2)^3$ means multiplying $5^2$ by itself three times:

5^2\cdot 5^2\cdot 5^2=5^{2+2+2}=5^6

Learn by doing: Apply the power-of-a-power rule

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

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Apply the power-of-a-power rule

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

Learn by doing: Apply the power-of-a-power rule

Practice with unlimited practice problems

Practice:

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