Squares, Cubes and Roots
Grade 3Square 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.
Worked example
Try this one
Without using a calculator, simplify fully \( \dfrac{4^{3} \times 8^{2}}{2^{10}} \).
Give your answer as a power of 2. [3]
This uses index laws with the same base.
Ask yourself: 'can I write every number in the expression as a power of 2?' Once all bases match, use the multiplication and division index laws: \( a^m \times a^n = a^{m+n} \) and \( a^m \div a^n = a^{m-n} \).
