Number - Structure and calculation: use positive integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 (Foundation tier)

The idea

An index tells you how many copies of the base are multiplied together, so 2⁵ means 2 × 2 × 2 × 2 × 2 = 32, and 4³ = 64. A root reverses that: the square root of 81 is 9, the cube root of 125 is 5, and the fourth root of 16 is 2. It pays to know the powers of two as far as 2¹⁰ = 1024, the powers of three as far as 3⁵ = 243, the powers of four as far as 4⁵ = 1024, and the powers of five as far as 5⁴ = 625, because spotting 243 as a power of three turns a hard question into a quick one. Every positive number has two square roots, one positive and one negative, and the √ sign written on its own means the positive one, so √49 = 7. Cube roots keep the sign of the number, so the cube root of −8 is −2.

What earns the marks

• Writing a power out as a repeated multiplication where that is the step being tested.
• The correct value of a power or a root, even where a later step then goes wrong.
• Recognising a given number as a power of a small base, such as writing 243 as 3⁵.
• Handling a power and a root in the same expression in the right order, indices before multiplying and dividing.

Common mistakes and how to put them right

• Reading 3⁴ as 3 × 4 and answering 12 rather than 81.

• Giving only the positive square root where a question about x² = 25 needs both 5 and −5.

• Confusing a cube root with dividing by three, so giving the cube root of 27 as 9.

• Assuming a square root is always smaller than the number, which fails between 0 and 1 where √0.25 = 0.5.

Exam tips

• Learn the squares as far as 15² and the cubes as far as 5³ so you recognise them on sight.
• On a calculator paper, find the power key and the root key before you need them and test one value you already know.
• Write a repeated multiplication out in full if you are unsure; it costs one line and stops you multiplying the base by the index.