Indices

1. 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.

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2. 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.

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3. 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 - ${\left(\frac{3}{4}\right)}^{-1}$ = $\frac{4}{3}$. The negative sign never makes the answer negative.

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4. 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.

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5. 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.

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• Multiplying the indices when multiplying powers, so $2^3$ × $2^5$ becomes $2^{15}$.

  With the same base, add the indices: $2^3$ × $2^5$ = $2^8$. Multiply them only for a power of a power, like (2³)⁴ = $2^{12}$.

• Treating a negative index as a negative number, so $2^{-3}$ becomes −8.

  A negative index means one over: $2^{-3}$ = $\frac{1}{2^3}$ = $\frac{1}{8}$. The answer stays positive.

• Writing $7^0$ as 0 or as 7.

  Any number except 0 to the power 0 is 1, so $7^0$ = 1. It fits the pattern: $7^2$ ÷ $7^2$ = $7^0$, and a number divided by itself is 1.

• Multiplying by a fractional index, so ${27}^{\frac{2}{3}}$ becomes 27 × $\frac{2}{3}$ = 18.Higher only

  The bottom of the fraction is a root and the top is a power. Cube-root 27 to get 3, then square it: ${27}^{\frac{2}{3}}$ = 9.

• Raising only the letter to the power, so (2x³)² becomes 2x⁶.

  Everything inside the bracket takes the power, the number too: (2x³)² = 2² × x⁶ = 4x⁶.

More Number topics

• Fractions
• Percentages
• Standard Form
• Surds
• Bounds & Rounding
• Factors & Multiples

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• The GCSE Maths syllabus, explained