Exponent rules (laws of exponents)

The exponent rules, also called the laws of exponents, or the index laws in the UK, are the shortcuts for working with powers. The three main ones: to multiply powers of the same base, add the exponents; to divide, subtract them; and to raise a power to a power, multiply them. Alongside those, any base to the power 0 is 1, a negative exponent means a reciprocal, and a fractional exponent means a root.

All the exponent rules in one chart

In aⁿ, a is the base and n is the exponent, also called the power or index. So 2⁵ means five 2s multiplied together: 2 × 2 × 2 × 2 × 2 = 32. Every rule below comes from that one idea.

The product and quotient rules only work when the bases are the same. x³ × y⁴ cannot be simplified to a single power.

The product rule: multiplying exponents with the same base

Write it out to see why: x³ × x⁴ is (x × x × x) × (x × x × x × x), which is seven x's, so x⁷. You add the exponents because you are counting how many times the base is multiplied in total.

With numbers in front, multiply the numbers and add the exponents: 3x² × 5x⁴ = 15x⁶.

The quotient rule: dividing exponents with the same base

x⁷ ÷ x² is seven x's on top and two underneath. Two pairs cancel, leaving five, so x⁵. With numbers, divide the numbers and subtract the exponents: 12x⁸ ÷ 4x³ = 3x⁵.

The power of a power rule

(x²)⁵ means x² multiplied by itself five times, which is 2 + 2 + 2 + 2 + 2 = 10 x's. So you multiply the exponents: (x²)⁵ = x¹⁰.

Power of a product and power of a quotient

A power outside a bracket applies to every factor inside:

So (2x³)⁴ = 2⁴ × x¹² = 16x¹². This only works for multiplying and dividing inside the bracket. (x + y)² is not x² + y²: it is x² + 2xy + y².

Try these first rules now. Simplify the expression, then press Mark to check your answer in the course.

The zero exponent rule

Use the quotient rule on x³ ÷ x³. The answer must be 1, since anything divided by itself is 1, but the rule gives x³⁻³ = x⁰. So x⁰ = 1. The pattern of powers of 2 shows the same thing: each step down divides by 2, so after 2¹ = 2 comes 2⁰ = 1.

Negative exponent rules

Keep dividing by 2 in the table and the exponents go negative: 2⁻¹ = 1/2 and 2⁻² = 1/4. A negative exponent does not make the answer negative. It means one over the positive power, the reciprocal.

• 5⁻² = 1/5² = 1/25
• x⁻⁴ = 1/x⁴
• 1/x⁻³ = x³: a negative power on the bottom moves to the top
• (2/3)⁻² = (3/2)² = 9/4: flip the fraction, then square it

Fractional exponent rules

A fractional exponent means a root. The denominator is which root, and the numerator is the power.

So 25^(1/2) = √25 = 5, and 27^(1/3) = ∛27 = 3. For a general fraction, take the root first while the number is small, then the power: 32^(3/5) = (⁵√32)³ = 2³ = 8.

The rule follows from the power of a power rule: (9^(1/2))² = 9¹ = 9, so 9^(1/2) must be the number that squares to 9, which is 3.

Negative fractional exponents

Combine the rules in order: the minus sign means flip, the denominator means root and the numerator means power.

Seeing the rules on a graph

The graph of y = 2ˣ joins up every whole, zero, negative and fractional power of 2. Here x is the exponent and y is the value of 2 to that power.

The curve passes through (0, 1) because 2⁰ = 1. To the left, negative exponents give fractions that get closer to 0 but never reach it. Halfway between 0 and 1, at x = ½, the curve is at √2 = 1.41: that is what a fractional exponent means.

Exponent rules for addition and subtraction

There is no rule that combines the powers when you add or subtract them. 2³ + 2⁴ is 8 + 16 = 24, not 2⁷ = 128. What you can do is collect like terms, which must have the same base and the same exponent: 3x² + 5x² = 8x². Or factor out a common power: 2³ + 2⁴ = 2³(1 + 2) = 24.

Worked examples

Example 1: simplify with several rules

Simplify (2a³b)⁴ ÷ 4a⁵b².

Apply the power to every factor in the bracket first: (2a³b)⁴ = 16a¹²b⁴. Then divide the numbers and subtract the exponents for each letter.

Example 2: evaluate a negative fractional power

Evaluate 16^(−3/4).

Flip it, take the fourth root, then cube: ⁴√16 = 2, and 2³ = 8. So 16^(−3/4) = 1/8.

Example 3: solve an equation with exponents

Solve 2^(x + 1) = 32.

Write 32 as a power of 2: 32 = 2⁵. If the bases are the same, the exponents must be equal.

So x = 4. Questions like this work when both sides can be written with the same base. When they cannot, you need logarithms.

Example 4: simplify a root using exponents

Simplify √(x⁶) × x^(1/2) as a single power of x.

√(x⁶) = (x⁶)^(1/2) = x³. Then x³ × x^(1/2) = x^(3 + 1/2) = x^(7/2).

Where you will use the exponent rules next

Scientific notation

Scientific notation (standard form in the UK) writes numbers as a × 10ⁿ, so the exponent rules do most of the work. To multiply, multiply the front numbers and add the powers of 10. To divide, divide the front numbers and subtract the powers.

• (3 × 10⁴) × (2 × 10⁵) = 6 × 10⁹
• (8 × 10⁶) ÷ (4 × 10²) = 2 × 10⁴
• (5 × 10³) × (4 × 10⁷) = 20 × 10¹⁰, which you then rewrite as 2 × 10¹¹

The Scientific Notation Calculator converts and multiplies numbers in this form.

Derivatives

In calculus, the power rule differentiates xⁿ to nxⁿ⁻¹. It only works on a single power of x, so you often use the exponent rules first: rewrite √x as x^(1/2) and 1/x² as x⁻², then differentiate. The Derivative Calculator shows this rewriting step by step.

Common mistakes

• Multiplying the bases. 2³ × 2⁴ = 2⁷, not 4⁷. The base stays the same.
• Multiplying exponents when you should add. x³ × x⁴ = x⁷, not x¹². Multiply only for a power of a power.
• Adding powers in a sum. x² + x³ cannot be simplified to x⁵.
• Thinking a negative exponent gives a negative number. 3⁻² = 1/9, which is positive.
• Forgetting the number in a bracket. (3x)² = 9x², not 3x².
• Mixing up −3² and (−3)². −3² = −9, because the square applies only to the 3. (−3)² = 9.

Are the exponent rules on the formula sheet?

No. These were checked in November 2026. The GCSE Maths formulae sheets from AQA, Edexcel and OCR do not include the index laws, and the reference sheet in College Board's official SAT practice tests lists only geometry formulas. You need to know these rules by heart for both.

Check answers like these in the Exponent Calculator, which works out powers and roots and shows the index law it uses.

Practise in a short timed test

Exponents run through Algebra 1 and Algebra 2, and through GCSE Maths. Pre-Algebra and Algebra 1 and Geometry, Algebra 2 and SAT Math each give you a short timed practice test every day, with 8 questions worth 25 to 35 points. In the UK, the same practice opens as GCSE Maths Higher. To build up from the basics, the Algebra course covers the exponent rules with an interactive exponent builder.

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