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 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.

On this page:

Learn “Apply the power-of-a-power rule” with a detailed explanation of the concept
Master “Apply the power-of-a-power rule” with unlimited, free practice problems
Download (or modify) a free worksheet on Apply the power-of-a-power rule you can use at home or in class with practice problems that scaffold and explore “Apply the power-of-a-power rule” from different perspectives.
Start working on adaptive units like “Exponents - Negative, Fractional, and Power Law” or “Exponents - Multiplication and Division - Advanced” that introduce, teach, or practice “Apply the power-of-a-power rule” to maximize your learning.

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

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Your results

Practice with unlimited practice problems

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

Earned ?

Download/Modify Free Grade 10 'Apply the power-of-a-power rule' Worksheet

Print this free 'Apply the power-of-a-power rule' worksheet to practice this skill away from the computer. Click 'Modify' to easily customize this worksheet to your exact needs.

Units Teaching This Skill

These adaptive units introduce, teach, or reinforce this skill (Apply the power-of-a-power rule) through scaffolded problems that grow in difficulty as concepts are understood.