Learn · Algebra and geometry (grades 8–10)

Exponent Rules: How to Simplify Powers, Step by Step

An exponent says how many times to multiply a base by itself: x³ = x · x · x. A few simple rules let you combine powers without writing them all out.

Try it with your own problem ↓

The steps

  1. Product rule: same base, multiply: add the exponents. xᵃ · xᵇ = xᵃ⁺ᵇ.
  2. Quotient rule: same base, divide: subtract. xᵃ ÷ xᵇ = xᵃ⁻ᵇ.
  3. Power rule: a power of a power: multiply the exponents. (xᵃ)ᵇ = xᵃᵇ. A power of a product applies to every factor.
  4. Zero and negative exponents: x⁰ = 1, and x⁻ⁿ = 1/xⁿ.
  5. Apply the rules one at a time, working from the inside out.

A worked example

Simplify (2x³)² · x⁴.

  1. Apply the square to every factor inside: (2x³)² = 2² · (x³)².
  2. 2² = 4, and (x³)² = x⁶ (multiply the exponents 3 × 2).
  3. Now we have 4x⁶ · x⁴.
  4. Product rule: x⁶ · x⁴ = x¹⁰ (add 6 + 4).
  5. Put it together: 4x¹⁰.

(2x³)² · x⁴ = 4x¹⁰.

Why it works

Each rule is just counting. x³ · x⁴ is (x · x · x) · (x · x · x · x), which is seven x's in a row, so x⁷. And (x³)² is x³ written twice, which is six x's, so x⁶. If you forget a rule, write the small case out in full.

Try it with your own problem

Open the free The Power Tower with this example already filled in. It shows every step, and you can type over it with your own numbers.

Work it out: (2x^3)^2*x^4Opens The Power Tower with this example

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Common mistakes

Questions people ask

Why is anything to the power 0 equal to 1?

From the quotient rule: x³ ÷ x³ = x⁰, and anything divided by itself is 1.

What is a negative exponent?

A reciprocal. 5⁻² = 1/5² = 1/25.

Do the rules work for fractional exponents?

Yes. x^(1/2) is the square root of x, and x^(2/3) is the cube root of x, squared.

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