EZ Math

Algebra 2 · Rational exponents and radicals

How to Raise a Power to a Power

Problem

(x3)4(x^3)^4

Answer

x12x^{12}

The power-of-a-power rule multiplies exponents. This is the opposite of the product rule, which adds exponents when the bases multiply.

Step-by-step solution

  1. Write the power-of-a-power rule.

    (xm)n=xmn(x^m)^n = x^{mn}

    Why: A power raised to a power repeats the base, so the exponents multiply.

  2. Multiply the exponents.

    3×4=123 \times 4 = 12

    Why: The outer exponent applies to the whole inner power.

  3. Write the simplified expression.

    x12x^{12}

    Why: The base x is unchanged. Only the exponent changes.

Common mistakes

  • Adding the exponents and writing x^7. Powers multiply inside a power.
  • Multiplying the base and writing 12x.
  • Leaving the answer as (x^3)^4. A simplified power has one exponent.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

  1. (x2)5(x^2)^5

    Show answer

    x10x^{10}

  2. (a4)3(a^4)^3

    Show answer

    a12a^{12}

  3. (y6)2(y^6)^2

    Show answer

    y12y^{12}

  4. (23)2(2^3)^2

    Show answer

    6464

  5. ((x2)3)2((x^2)^3)^2

    Show answer

    x12x^{12}