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Math · Original worked example

Choose the exponent rule that matches the operation

Multiply powers, raise a power to a power, and interpret negative exponents with an original worked example and a fresh ACT-style question.

By Raise Your ACT · Updated October 27, 2026 · How we review content

Before you begin

Useful foundation: Understand a power as repeated multiplication and recognize a nonzero denominator.

Your goal: Distinguish adding exponents from multiplying them, and rewrite a negative exponent as a reciprocal.

For the same nonzero base, multiply powers by adding exponents and divide powers by subtracting them. When raising a power to a power, multiply the exponents. A negative exponent means a reciprocal, not a negative value.

Work through an example

For x ≠ 0, simplify (x³)² ÷ x⁴.
  1. A power of a power multiplies exponents: (x³)² = x⁶. You can also see this as x³ × x³.
  2. Divide powers with the same base by subtracting exponents: x⁶ ÷ x⁴ = x².
  3. The original denominator is x⁴, so x must not be zero even though the simplified expression x² can be evaluated at zero.
  4. Check with x = 2: (2³)² ÷ 2⁴ = 64 ÷ 16 = 4, which equals 2².

Try a question yourself

For y ≠ 0, which expression is equivalent to (y²)³ × y⁻⁴?

Choose an answer, then check your reasoning. This public practice is not saved.

Read all answer explanations

y⁻¹⁰: The power of a power gives y⁶. Multiplying by y⁻⁴ adds 6 and −4, rather than making both exponents negative.

y⁻²: Subtracting 6 from 4 reverses the operation. The expression is y⁶ × y⁻⁴, so the exponent is 6 + (−4) = 2.

y² — correct answer: Yes. (y²)³ = y⁶, and y⁶ × y⁻⁴ = y² for nonzero y. A negative exponent contributes a reciprocal.

−y²: A negative exponent does not put a minus sign in front of the value. For example, y = 2 gives 64 × 1/16 = 4, not −4.

What to do next

Name the operation before using a rule. Try y = 2 as a numerical check, then explain why the original restriction y ≠ 0 remains.

Revisit the idea in a week using a different question. Repeating this example can help you understand it, but remembering its answer is not evidence of learning a new problem.

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These original practice examples are independent of ACT and are not official test questions.