Squares, Cubes and Roots

Grade 3

Square numbers — what they are

A square number is what you get when you multiply a whole number by itself. So 3 × 3 = 9, meaning 9 is a square number. We write this as 3² = 9 and say "three squared equals nine".

The first few square numbers are worth knowing off the top of your head: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Recognising them quickly saves real time in exams.

Cube numbers — same idea, one step further

A cube number is a whole number multiplied by itself three times. So 2 × 2 × 2 = 8, meaning 8 is a cube number. We write this as 2³ = 8 and say "two cubed equals eight".

The cube numbers to know: 1, 8, 27, 64, 125, 216. Notice that 64 appears in both lists: 8² = 64 (so it's a square number) and 4³ = 64 (so it's a cube number too). 64 is both.

Roots — going backwards

If squaring a number is the operation, then the square root (written √) is its inverse — it undoes the squaring. So √49 = 7, because 7² = 49.

The cube root (written ∛) undoes cubing in the same way. ∛27 = 3, because 3³ = 27.

Roots and powers are inverses of each other. If you know your square and cube numbers, you already know their roots — just read the list backwards.

Worked example

EXAMPLE

Without a calculator, find: (a) √144 (b) ∛125

(a) Think: which number multiplied by itself gives 144? 12 × 12 = 144, so √144 = 12. (b) Think: which number cubed gives 125? 5 × 5 × 5 = 125, so ∛125 = 5.

A common slip: √16 = 4, not 8. The square root of 16 is 4 (because 4 × 4 = 16), not half of 16. Halving and square-rooting are different operations.

Try this one

Work out the value of \( 4^3 - 2^4 \). Show your working.

[3]

Nova's hint:

This involves evaluating two different powers and then subtracting.

Ask yourself: 'what does each power mean in terms of repeated multiplication?' Work out each term separately before subtracting. For example, \( 3^2 = 9 \) and \( 2^3 = 8 \).

Try to Solve it with Nova?