1.5 Squares, Cubes and Roots

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: square numbers, cube numbers, square roots and cube roots, and estimating roots. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Recall the square numbers up to 15 × 15 and the cubes of 1, 2, 3, 4, 5 and 10
• Find square roots and cube roots of square and cube numbers
• Estimate the square root of a number that is not a square number
• Use squares, cubes and roots in calculations, including with a calculator

Key words

• Square number: A number multiplied by itself. 7 × 7 = 7² = 49, so 49 is a square number.
• Cube number: A number used three times in a multiplication. 4 × 4 × 4 = 4³ = 64.
• Square root: √ undoes squaring. √49 = 7 because 7² = 49.
• Cube root: ∛ undoes cubing. ∛64 = 4 because 4³ = 64.

Square numbers and cube numbers

Square numbers make square patterns. Cube numbers make cubes.

• n²; 1 1; 2 4; 3 9; 4 16; 5 25; 6 36; 7 49; 8 64; 9 81; 10 100; 11 121; 12 144; 13 169; 14 196; 15 225
• n³; 1 1; 2 8; 3 27; 4 64; 5 125; 6 ; 7 ; 8 ; 9 ; 10 1000; 11 ; 12 ; 13 ; 14 ; 15

Worked example: squares and cubes together

Work out 7² + 3² and 2³ × 5
7² + 3² = 49 + 9 = 58
2³ × 5 = 8 × 5 = 40

Answer: 58 and 40

Square roots

Every positive number has two square roots, one positive and one negative: 6 × 6 = 36 and −6 × −6 = 36. The √ sign means the positive one, so √36 = 6.

Worked example: roots from the table

(a) √121 = 11 (b) √196 = 14
(c) Solve x² = 81: x = 9 or x = −9

Answer: 11, 14, ±9

To estimate a square root, find the square numbers either side.

Cube roots

A cube root undoes cubing. A negative number has a negative cube root, because (−3)³ = −3 × −3 × −3 = −27.

Worked example: cube roots

(a) ∛125 = 5 (b) ∛1000 = 10 (c) ∛(−27) = −3

Answer: 5, 10, −3

Worked example: a cube box

A cube-shaped box has a volume of 64 cm³. How long is each edge?
Volume = edge × edge × edge, so edge = ∛64 = 4 cm

Answer: 4 cm

Calculations with powers and roots

Powers and roots come before ×, ÷, + and − in the order of operations. A root sign over a sum acts like a bracket: work out what is under it first.

Worked example: roots and brackets

(a) √(9 + 16) = √25 = 5
(b) √9 + √16 = 3 + 4 = 7
These are different! You cannot split a root over a + sign.

Answer: (a) 5 (b) 7

Worked example: with a calculator

(a) √20.25 = 4.5 (b) 3.2³ = 32.768
(c) √(7² + 24²) = √(49 + 576) = √625 = 25

Answer: 4.5, 32.768, 25

Tip: On a calculator use the x², x³, √ and ∛ keys. Put brackets round anything under a root sign.

Common mistakes

• Doubling instead of squaring: 6² = 36, not 12.
• Multiplying by 3 instead of cubing: 4³ = 64, not 12.
• Splitting a root: √(16 + 9) is √25 = 5, not √16 + √9 = 7.
• Saying √50 ≈ 25 by halving. Use the square numbers either side: √50 is just over 7.

Revision questions

Try these, then check your answers below each one.

Question 1. Work out 12².

Answer: 144

Working: 12 × 12 = 144.

Question 2. Work out 3√64.

Answer: 4

Working: 4 × 4 × 4 = 64, so 3√64 = 4.

Question 3. √90 lies between which two whole numbers? (Choose from: 8 and 9, 9 and 10, 10 and 11, 45 and 46.)

Answer: 9 and 10

Working: 9² = 81 and 10² = 100. 90 is between them, so √90 is between 9 and 10.

Related lessons

• Previous lesson: 1.4 Factors and Multiples
• Next lesson: 1.6 Index Notation
• In this chapter: 1.1 Calculations
• In this chapter: 1.2 Decimal Numbers
• In this chapter: 1.3 Place Value

Continue in the full course

This lesson is section 1.5 (EGCMF0105) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.