What this chapter is about
This chapter introduces you to two important ideas in mathematics: square numbers and cube numbers. You already know how to multiply numbers. When you multiply a number by itself, you get its square. When you multiply a number by itself twice (three copies of the number multiplied together), you get its cube.
Understanding squares and cubes helps you in many areas — calculating areas of square plots, finding volumes of cube-shaped boxes, and working with algebraic expressions later. These concepts also connect to finding square roots and cube roots, which means working backwards to find what number was squared or cubed.
After studying this chapter, you should be able to find squares and cubes of numbers, recognise patterns in square and cube numbers, and calculate square roots and cube roots of perfect squares and perfect cubes.
Key ideas
- A square number is the result of multiplying a whole number by itself. For example, 5 × 5 = 25, so 25 is the square of 5, written as 5² = 25.
- A cube number is the result of multiplying a whole number by itself twice. For example, 4 × 4 × 4 = 64, so 64 is the cube of 4, written as 4³ = 64.
- The square root of a number is the value that, when multiplied by itself, gives that number. The square root of 36 is 6, because 6 × 6 = 36.
- The cube root of a number is the value that, when multiplied by itself twice, gives that number. The cube root of 27 is 3, because 3 × 3 × 3 = 27.
- Square numbers are always positive. The squares of both positive and negative numbers give positive results: (−3)² = 9 and 3² = 9.
- Cube numbers keep the sign of the original number: (−2)³ = −8 while 2³ = 8.
- Not every number is a perfect square or perfect cube. For instance, 20 is not a perfect square, and 50 is not a perfect cube.
Formulas and facts to remember
- Square of n: n² = n × n (area of a square with side n units)
- Cube of n: n³ = n × n × n (volume of a cube with side n units)
- Square root of n²: √(n²) = n
- Cube root of n³: ∛(n³) = n
- First ten perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
- First ten perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
- Pattern: The difference between consecutive squares increases by 2 each time (4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7, and so on)
Worked examples
Example 1: Finding the square and cube of a number
Find the square and cube of 7.
Solution:
- Square of 7 = 7² = 7 × 7 = 49
- Cube of 7 = 7³ = 7 × 7 × 7 = 49 × 7 = 343
So, 7² = 49 and 7³ = 343.
Example 2: Finding square root by prime factorisation
Find the square root of 144.
Solution: First, write 144 as a product of prime factors: 144 = 2 × 72 = 2 × 2 × 36 = 2 × 2 × 2 × 18 = 2 × 2 × 2 × 2 × 9 = 2 × 2 × 2 × 2 × 3 × 3
So, 144 = 2⁴ × 3²
For square root, take half the power of each prime: √144 = 2² × 3¹ = 4 × 3 = 12
Therefore, √144 = 12.
Example 3: Finding cube root
A water tank shaped like a cube holds 512 litres. If 1 litre fills a cube of side 10 cm, what is the side length of the tank in decimetres?
Solution: Volume of tank = 512 cubic decimetres (since 1 litre = 1 cubic decimetre)
Side of cube = cube root of 512 = ∛512
Finding by factorisation: 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁹
For cube root, take one-third of the power: ∛512 = 2³ = 8
The side of the tank is 8 decimetres (or 80 cm).
Common mistakes
- Confusing square with doubling → 5² means 5 × 5 = 25, not 5 × 2 = 10.
- Forgetting that squaring a negative number gives a positive result → (−4)² = 16, not −16.
- Thinking cube root of a negative number does not exist → (−3)³ = −27, so ∛(−27) = −3.
- Missing pairs during prime factorisation → always group factors in pairs for square root, in triplets for cube root.
- Writing √25 = ±5 in basic problems → the principal square root is the positive value, so √25 = 5.
Quick revision
- n² means n times n; n³ means n times n times n.
- Square root reverses squaring; cube root reverses cubing.
- Use prime factorisation: pair factors for square root, group in threes for cube root.
- Squares are always positive; cubes carry the sign of the original number.
- Memorise squares up to 15 and cubes up to 10 for quick calculations.