A Square and A Cube

Overview

This chapter explores square numbers and cube numbers, their patterns and properties. A square number (perfect square) is obtained by multiplying a number by itself, and a cube number by multiplying it three times. We also learn to find square roots and cube roots.

Key concepts

• Square of n is n × n, written n²; cube of n is n × n × n, written n³.
• Perfect squares end in 0, 1, 4, 5, 6 or 9 — never in 2, 3, 7 or 8.
• The sum of the first n odd numbers equals n² (e.g. 1 + 3 + 5 = 9 = 3²).
• Square root (√) and cube root (∛) reverse these operations.
• Prime factorisation helps find roots: pair factors for square roots, group in threes for cube roots.

Important formulae

• (a + b)² = a² + 2ab + b²
• n² − (n−1)² = 2n − 1
• √(a × b) = √a × √b

Solved example

1. Find √1296 by prime factorisation.
2. 1296 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3.
3. Pair them: (2×2)(2×2)(3×3)(3×3).
4. Take one from each pair: 2 × 2 × 3 × 3 = 36. So √1296 = 36.

Important questions

1. Find the smallest number by which 252 must be multiplied to make it a perfect square.
2. Find the cube root of 3375 by prime factorisation.
3. Without calculating, state whether 1057 can be a perfect square. Give a reason.
4. How many natural numbers lie between 12² and 13²?

Quick revision

Squares grow by consecutive odd numbers; cubes grow faster. Use prime factorisation, pairing for square roots and tripling for cube roots. Remember the last-digit test to reject non-squares quickly.