GCSE Maths · Topic guide
Indices
Indices (or powers) are a shorthand for repeated multiplication: a^n means a multiplied by itself n times. The rules of indices let you simplify expressions involving powers without expanding them out, which is essential for algebra, standard form and surds.
Before you start
Make sure you're comfortable with these topics first:
Powers and RootsFactors and Multiples
Method
- When multiplying powers of the same base, add the indices: a^m x a^n = a^(m+n).
- When dividing powers of the same base, subtract the indices: a^m divided by a^n = a^(m-n).
- When raising a power to another power, multiply the indices: (a^m)^n = a^(mn).
- Any non-zero number to the power 0 equals 1: a^0 = 1.
- A negative index means 'reciprocal': a^-n = 1/(a^n).
- A fractional index means a root: a^(1/n) is the nth root of a, and a^(m/n) is the nth root of a, raised to the power m.
Worked example
Simplify (3x^4)^2 x 2x^-3, giving your answer as a single term.
- Deal with the bracket first: (3x^4)^2 = 3^2 x x^(4x2) = 9x^8.
- Multiply by 2x^-3: 9x^8 x 2x^-3 = (9 x 2) x x^(8 + -3).
- 9 x 2 = 18, and 8 + -3 = 5.
- So the simplified expression is 18x^5.
Practice questions
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Exam-style questions
Written in the style of a GCSE exam paper, with a full mark scheme.
Q1[2 marks]
Simplify fully: (x^6 x x^2) divided by x^3.
Q2[3 marks]
Work out the value of 8^(2/3). Show your method.
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