KS3 · Maths

Square and cube numbers and their roots

64 is a square number AND a cube number. 48 is neither, even though 4 × 4 × 3 looks tempting. Let's sort out why.

Number

Square, cube, both… or neither?

For each number, hunt for the repeated whole number. A square number is a whole number times itself (5 × 5 = 25). A cube number is a whole number times itself, then times itself again (3 × 3 × 3 = 27). Tick every set each number belongs in. Tick none if you think it's in neither.

  • A Square numbers
  • B Cube numbers
  1. 4
  2. 9
  3. 144
  4. 8
  5. 27
  6. 216
  7. 1
  8. 64
  9. 3
  10. 48
  11. 80
  12. 12.25

Undo it with a root

Problem

A cube-shaped box has a volume of 512 cm³. How long is each edge?

Spot the slip

Where does this answer go wrong?

Work out ∛64 − √64.

A student's answer — which line goes wrong?

Predict, then test

Use a calculator for the test if you want to.

A root that isn't a whole number still sits between two whole numbers. √20 does: 4 × 4 = 16 and 5 × 5 = 25, and 20 is between them, so √20 is between 4 and 5. You can write it exactly as √20, or approximately as a decimal (about 4.47). Now ∛92. Since 4 × 4 × 4 = 64 and 5 × 5 × 5 = 125, it is between 4 and 5 too. 92 is nearer to 64 than to 125. To decide which whole number ∛92 is closer to, compare 92 with the cube of the halfway number 4.5. Predict first: is ∛92 closer to 4 or to 5?

Use it

Write it out

Mia wants a square patch of grass with whole-number side lengths in metres. Its area must not exceed 20 m². What is the side length of the largest patch she can have? Explain how you know. [3 marks]

0 words · your answer stays on this page and is not sent anywhere.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Spot them, undo them, and find out where a root lands when it isn't a whole number.

What you need to know

  • A square number is a whole number multiplied by itself, like 5 × 5 = 25 (also called a perfect square). A cube number is a whole number multiplied by itself twice, like 3 × 3 × 3 = 27.
  • 1 and 64 are both square and cube numbers. Many numbers, like 48, 80 and 12.25, are neither.
  • √ undoes squaring and ∛ undoes cubing. Check a root by multiplying back. A root is never just half the number.
  • A root that isn't a whole number lies between two neighbouring whole numbers. You can write it exactly (√20) or approximately as a decimal.
  • To decide which whole number a root is closer to, compare the number with the square or cube of the halfway value: 4.5³ = 91.125 tells you ∛92 is closer to 5.

The big picture

A square number is a whole number times itself (n × n) and a cube number is a whole number times itself twice (n × n × n). The square root √ and cube root ∛ undo them. When a root isn't a whole number, find the two whole numbers it sits between, then compare with the square or cube of the halfway value to see which one it is closer to.

Key points

1Square number = n × n. Cube number = n × n × n, with n a whole number.
2√ undoes a square and ∛ undoes a cube: if 6 × 6 = 36 then √36 = 6.
3Trial and improvement: test, compare, adjust, repeat.
4A root between two whole numbers: use the halfway test (4.5³ = 91.125, 5.5³ = 166.375, 4.5² = 20.25) to say which whole number it is closer to.

Worked example

Problem

Work out √169 + ∛125.

⚠ Watch out

Writing √36 = 18 because 18 is half of 36. A square root undoes squaring: 6 × 6 = 36, so √36 = 6. Check any root by multiplying it back.

🧠

Memory hook

A root is a rewind button: √ rewinds a square, ∛ rewinds a cube. Rewinding never means halving.

✓

Check yourself

Is √70 closer to 8 or 9? Test it with 8.5² = 72.25. Answer: 70 is smaller, so closer to 8.

Flashcards

(12)
What is a square number?
A whole number multiplied by itself, like 5 × 5 = 25. Also called a perfect square.
What is a cube number?
A whole number multiplied by itself, then by itself again, like 3 × 3 × 3 = 27.
Which numbers are both square and cube numbers?
1 (1 × 1 and 1 × 1 × 1) and 64 (8 × 8 and 4 × 4 × 4).
Can a number be neither a square nor a cube?
Yes. 3, 48, 80 and 12.25 are all in neither set. 48 = 4 × 4 × 3 is a cuboid, not a cube.
What do √ and ∛ do?
√ (square root) undoes squaring: √25 = 5 because 5 × 5 = 25. ∛ (cube root) undoes cubing: ∛27 = 3 because 3 × 3 × 3 = 27.
How do you check a root you've found?
Multiply it back: a square root times itself, or a cube root times itself twice, should give the number you started with.
Is √36 equal to 18?
No. A root is not half the number. √36 = 6 because 6 × 6 = 36.
How does trial and improvement work for a root?
Test a whole number, compare the result with the target, adjust up or down, and repeat.
Where does √20 lie?
Between 4 and 5, because 16 < 20 < 25.
How can you write a root that isn't a whole number?
Exactly, like √20, or approximately as a decimal.
How do you decide whether ∛92 is closer to 4 or to 5?
Cube the halfway number: 4.5³ = 91.125. 92 is bigger than that, so ∛92 is closer to 5.
Is √20 closer to 4 or to 5?
Closer to 4. The halfway test: 4.5² = 20.25, and 20 is smaller than that.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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