Index laws, explained with examples

By The STEM-Tutor team · Updated October 2026

Indices (powers) are a shorthand for repeated multiplication. A small set of index laws lets you simplify expressions, handle negative and fractional powers, and solve equations, and they are used throughout GCSE and A level Maths.

What an index means

In aⁿ, a is the base and n is the index (or power, or exponent). For a positive whole number n, aⁿ means n copies of a multiplied together: 2⁴ = 2 × 2 × 2 × 2 = 16. Every index law below comes from this simple idea, so if you forget a rule, you can rebuild it by writing out the multiplication.

The laws

• Multiplying: a^m × aⁿ = a^m+n (add the powers).
• Dividing: a^m ÷ aⁿ = a^m−n (subtract the powers).
• Power of a power: (a^m)ⁿ = a^mn (multiply the powers).
• Zero index: a⁰ = 1, for any non-zero a.
• Negative index: a⁻ⁿ = 1/aⁿ (take the reciprocal).
• Fractional index: a^1/n = ⁿ√a, and a^m/n = (ⁿ√a)^m.

The multiplying and dividing laws only work when the bases are the same. 2³ × 3² cannot be simplified to a single power.

Why the laws work

2³ × 2⁴ = (2 × 2 × 2) × (2 × 2 × 2 × 2), which is seven 2s multiplied: 2⁷, so the powers add. Dividing cancels factors: 2⁵ ÷ 2² leaves three 2s, so the powers subtract. Following the dividing law, 2³ ÷ 2³ = 2⁰, but anything divided by itself is 1, which is why a⁰ = 1. And 2² ÷ 2⁵ = 2⁻³, which must equal 1/2³, explaining negative indices.

For fractional indices, notice that (9^1/2)² = 9¹ = 9 by the power law. The number that squares to 9 is 3, so 9^1/2 = √9 = 3.

Worked examples: numbers

Example 1: the basic laws

Write as a single power: 5³ × 5⁶, 7⁹ ÷ 7⁴, and (3²)⁵.

• 5³⁺⁶ = 5⁹.
• 7⁹⁻⁴ = 7⁵.
• 3^2×5 = 3¹⁰.

Example 2: zero and negative powers

Evaluate 6⁰, 2⁻³ and (2/5)⁻².

• 6⁰ = 1.
• 2⁻³ = 1/2³ = 1/8.
• (2/5)⁻² = (5/2)² = 25/4. A negative power of a fraction flips the fraction.

Example 3: fractional powers

Evaluate 25^1/2, 27^1/3, 8^2/3 and 16^−3/4.

• 25^1/2 = √25 = 5. 27^1/3 = ∛27 = 3.
• 8^2/3 = (∛8)² = 2² = 4. Take the root first; the numbers stay small.
• 16^−3/4 = 1/(⁴√16)³ = 1/2³ = 1/8. Deal with the negative (reciprocal), the denominator (root) and the numerator (power) one at a time.

Worked examples: algebra

Example 4: simplifying

Simplify 3x⁴ × 5x².

• Multiply the numbers and add the powers of x: 15x⁶.
• 15x⁶.

Example 5: dividing

Simplify 20a⁷b³ ÷ 4a²b.

• Divide the numbers: 5. Subtract powers: a⁵, b².
• 5a⁵b².

Example 6: a power of a product

Simplify (2x³y)⁴.

• Every factor inside the bracket is raised to the power 4: 2⁴x¹²y⁴.
• 16x¹²y⁴. Remember to raise the number to the power too.

Example 7: mixed

Simplify (x⁵ × x⁻²) ÷ x⁴.

• x^{5 + (−2)} = x³. Then x³⁻⁴ = x⁻¹.
• x⁻¹ = 1/x.

Calculator and non-calculator tips

Many index questions appear on the non-calculator paper, so learn the small powers by heart: squares to 15², cubes to 5³ or 10³, and powers of 2 up to 2¹⁰ = 1024. On a calculator, use the x^■ or ^ key, and put negative or fractional powers in brackets, such as 8^(2/3), or the calculator may work out 8² ÷ 3 instead. You can always check an index law with small numbers: if you think x³ × x⁴ = x¹², try x = 2 and see that 8 × 16 = 128 = 2⁷, not 2¹².

Solving index equations (Higher)

To solve an equation with the unknown in the power, write both sides as powers of the same base, then set the powers equal.

Example 8: same base

Solve 2^x = 32 and 9^x = 27.

• 32 = 2⁵, so x = 5.
• Write both as powers of 3: 9^x = 3^2x and 27 = 3³. So 2x = 3 and x = 3/2.

Example 9: a negative answer

Solve 4^x = 1/16.

• 1/16 = 4⁻².
• x = −2.

Indices, surds and standard form

Index laws connect to other GCSE topics. Surds are fractional powers: √x = x^1/2, so √x × √x = x¹ = x. Standard form uses powers of 10, and multiplying numbers in standard form is just the multiplying law: 10⁴ × 10⁵ = 10⁹. At A level, the same laws are essential for differentiation and integration, where 1/x² is rewritten as x⁻² before applying the power rule.

Common mistakes

• Multiplying the powers when multiplying terms: x³ × x⁴ = x⁷, not x¹².
• Thinking a negative index makes the answer negative: 2⁻³ = 1/8, not −8.
• Saying a⁰ = 0; it equals 1.
• Forgetting to raise the number in front to the power, as in (2x)³ = 8x³.
• Applying index laws to different bases.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Are index laws on the Foundation paper?

Yes. Multiplying, dividing and powers of powers, plus zero indices, are on both tiers. Negative and fractional indices are mainly Higher.

What is the difference between index, power and exponent?

They mean the same thing. UK textbooks usually say index or power; US textbooks often say exponent.

Key takeaways

• Multiply: add powers. Divide: subtract powers. Power of a power: multiply.
• a⁰ = 1 and a⁻ⁿ = 1/aⁿ.
• a^1/n is the nth root; a^m/n = (nth root)^m.
• Laws only combine powers with the same base.
• Solve index equations by matching bases.

Related guides

• GCSE Maths surds
• Standard form
• Logarithms