GCSE Maths · Number

Indices

Laws of indices, and (Higher) negative and fractional indices.

199 GCSE-style practice questions

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Indices, explained point by point

Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.

Use index notation and evaluate powers and roots without a calculator
The index counts how many copies are multiplied: 3⁴ = 3 × 3 × 3 × 3 = 81. Roots undo powers: $\sqrt{81}$ = 9 and the cube root of 64 is 4. Knowing the square numbers to 15² and cubes to 5³ makes non-calculator papers much friendlier.
Use the laws of indices to multiply and divide powers, and raise a power to a power
With the same base: ADD indices to multiply (2³ × 2⁵ = 2⁸), SUBTRACT to divide (2⁷ ÷ 2⁴ = 2³), and MULTIPLY for a power of a power ((2³)⁴ = 2¹²). The laws only work when the bases match - 2³ × 5² cannot be combined.
Work with zero and negative indices
Anything (except 0) to the power 0 is 1: 7⁰ = 1. A negative index means "one over": 2⁻³ = $\frac{1}{2^{3}}$ = $\frac{1}{8}$ , and with fractions it flips the whole thing - $\frac{3}{4}$ ⁻¹ = $\frac{4}{3}$ . The negative sign never makes the answer negative.
Work with fractional indicesHigher only
A fractional index is a root: $25^{\frac{1}{2}}$ = $\sqrt{25}$ = 5 and $8^{\frac{1}{3}}$ = 2. With a fraction like $\frac{2}{3}$ , the bottom is the root and the top is the power: $27^{\frac{2}{3}}$ means cube-root 27 first (getting 3), then square it - 9. Root first keeps the numbers small.
Combine the index laws in algebraic expressions
The same laws drive algebra: x³ × x⁵ = x⁸ and (2x³)² = 4x⁶ - square the 2 as well as the x³. Simplifying something like 12x⁷ ÷ 4x² means dividing the numbers and subtracting the indices: 3x⁵.

Practice

Try a Indices question

A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.

Simplify 2x³ × 5x⁴.

10x¹²
7x¹²
10x⁷
7x⁷

Show the answer and the working

Answer: 10x⁷

Multiply the coefficients (2 × 5 = 10) and add the indices of x (3 + 4 = 7).

Deal with the numbers and the x parts separately.
Coefficients: 2 × 5 = 10.
Powers of x: add the indices, 3 + 4 = 7.
Combine to get 10x⁷.

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Indices, explained point by point

Use index notation and evaluate powers and roots without a calculator

Use the laws of indices to multiply and divide powers, and raise a power to a power

Work with zero and negative indices

Work with fractional indicesHigher only

Combine the index laws in algebraic expressions

Try a Indices question

Make Indices stick

Ready to practise Indices?

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