Powers and roots

• Know the square numbers to 15² and the cubes 1, 8, 27, 64, 125 and 1000; find squares, roots, cubes and cube roots.
• Index laws: a^m × aⁿ = a^m+n, a^m ÷ aⁿ = a^m−n, (a^m)ⁿ = a^mn, a⁰ = 1, a⁻ⁿ = 1/aⁿ.
• Write an integer as a product of powers of prime factors, e.g. 720 = 2⁴ × 3² × 5.
• Use prime factors to find the HCF and LCM.
• Higher: simplify and manipulate surds, rationalise denominators, and evaluate fractional and negative powers.

Learn the lists

A square number is n × n; a cube number is n × n × n. The square root undoes squaring, the cube root undoes cubing. A positive number has two square roots (√49 means the positive one, 7, but x² = 49 gives x = ±7). A negative number has a negative cube root: ∛(−27) = −3.

Positive, negative and zero integer powers

In aⁿ, a is the base and n is the index (power). The laws only work when the bases are the same. None of these is on the formulae sheet; they must be memorised.

Prime factor form with indices

Break the number into primes (factor tree or repeated division), then collect repeated primes as powers, in order of size.

Using prime factor form

Write both numbers in prime factor form.

Exact roots, simplified

A surd is a root that is not a whole number, such as √2 or √12. Surds are exact; a calculator decimal is not. Simplify a surd by taking out the largest square factor.