IGCSE Maths / Powers and roots
Squares, cubes and their roots revision notes
Edexcel International GCSE Maths (4MA1), chapter 3: Powers and roots, specification point 1.4a-f. Free lesson.
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Squares and cubes are powers made by multiplying a number by itself, while roots undo those operations. Knowing common squares and cubes helps you calculate accurately and work more quickly without a calculator.
What squaring, cubing and roots mean
When you square a number, you multiply it by itself. When you cube it, you multiply three copies together. A square root or cube root undoes the matching power. 5² = 5 × 5 = 25 3³ = 3 × 3 × 3 = 27 √49 = 7, because 7 × 7 = 49
Which squares and cubes to know
You are expected to recognise the squares up to 15² and the cubes up to 5³ without a calculator. These facts also help when simplifying surds and factorising. Squares: 1, 4, 9, 16, 25; then 36, 49, 64, 81, 100; then 121, 144, 169, 196, 225. Cubes: 1, 8, 27, 64, 125; then 216, 343, 512, 729, 1000. 64 is in both lists: 8² = 64 and 4³ = 64.
How to work out a mixed calculation
Work out each power or root first, then do the remaining calculation. For a cube, multiply three copies of the number. For a square root, find the number that multiplies by itself to give the value inside the root.
Why a square has two roots
A positive number has two square roots, one positive and one negative. This is because multiplying two negative numbers gives a positive result. The square root sign on its own means the positive root, but an equation involving a square can have both answers. 7² = 49 and (−7)² = 49. So 49 has square roots 7 and −7, while √49 means 7.
How to avoid confusing a square with doubling
The small number written as a power tells you how many copies to multiply together. It is not a number to multiply the base by. 5² means 5 × 5 = 25, not 5 × 2 = 10. 3³ means 3 × 3 × 3 = 27, not 3 × 3 = 9.
Practise roots and powers together
For a calculation with different operations, work out each root or power before adding. Keep track of which number is being cubed and which is being squared.