One more direction

A square number fills a flat square. A cube number fills a solid cube. Eight small blocks stack into a 2 by 2 by 2 cube, and twenty seven make a 3 by 3 by 3 one.

Cubing and cube rooting

Operation	What it does	Example
Cubing	Multiply a number by itself three times	4³ = 4 × 4 × 4 = 64
Cube rooting	Find the number that was cubed	cube root of 64 is 4

Three times, not three lots

4 cubed is 4 times 4 times 4, which is 64. It is not 4 times 3, and it is not 4 plus 4 plus 4. The small 3 counts how many copies are being multiplied, not added.

The cube numbers

The first ten cubes are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000. They grow far faster than the squares: by the time you reach 10, the square is 100 but the cube is already 1000.

Why cubes grow so fast

Double the side of a square and the area becomes four times bigger. Double the side of a cube and the volume becomes eight times bigger. Lengths scale by k, areas by k squared, volumes by k cubed.

A useful difference from squares

Cubing 0 to 9 gives last digits 0, 1, 8, 7, 4, 5, 6, 3, 2, 9. Every digit appears exactly once, so unlike squares, the last digit never rules a cube out. Instead it tells you exactly what the last digit of the root must be.

The two block method for a cube root

Split a six digit number into blocks of three from the right: 262 and 144. The 144 ends in 4, and only 8 cubed ends in 4, so the root ends in 8. The 262 sits between 6 cubed and 7 cubed, so the root begins with 6. The answer is 68.

Cube roots by prime factorisation

The method is the same as for square roots with one change: group the prime factors into threes instead of pairs, and take one factor from each group of three.

Cube roots have no sign problem

Minus 2 cubed is minus 8, because three negative signs multiplied leave a negative. So every number has exactly one cube root, and the cube root of a negative number is negative.

🔐

The full lesson is waiting for you

Create a free account to unlock activities, practice tests, presentations and videos - and to track your progress and confidence score for every subject.

Create free account Sign in

Worked Example 1

Find the cube root of 1728 by prime factorisation.

Step 1: break it into primes1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3

Step 2: group in THREES(2 × 2 × 2), (2 × 2 × 2), (3 × 3 × 3)

Step 3: take one from each group2 × 2 × 3 = 12

Check both ways12 × 12 × 12 = 1728. And 1728 ends in 8, so the root must end in 2, which it does.

Worked Example 2

Find the cube root of 262144 using the two block method.

Split into blocks of three from the right262144 becomes 262 and 144.

The right block gives the last digit144 ends in 4, and only 8 cubed ends in 4, so the root ends in 8.

The left block gives the first digit262 lies between 6³ and 7³, so the root begins with 6.

Put them togetherThe cube root of 262144 is 68, far quicker than factorising a six digit number.

🔐

1 more worked examples - sign in to see them all.

Create free account Sign in

Activity 12.2

Build cubes from blocks and find how many you need each time

What you needSixty or more identical small cubes: sugar lumps, dice, small blocks, or paper cubes folded from nets.

MethodBuild 1 by 1 by 1, then 2 by 2 by 2, then 3 by 3 by 3, counting the blocks each time.

The question to answerAlongside the cube count, record the square number for the same side. Compare how differently the two columns grow.

🔐

That's a peek at activity 1 of 4 - sign in for all of them, full length.

Create free account Sign in

Cubing means multiplying a number by itself three times: 4³ = 4 × 4 × 4 = 64.

🔐

12 more points to remember - sign in to see the rest.

Create free account Sign in

1What is a cube number, and how does the picture behind the word differ from a square number?

2Explain how to find the cube root of 1728 by prime factorisation.

3Why does the last digit of a number tell you the last digit of its cube root, when the same trick does not work for square roots?

4A water tank is doubled in every direction. How much more water does it hold, and why?

5Why does every number have a cube root while negative numbers have no square root?

Question 1 of 10

1What is 4 cubed?

🔐

Create free account Sign in

Unit 12 · Cubes and Cube Roots

Cubes and Cube Roots

Mathematics · Grade 7

What You Will Be Able To Do

🧊

Cube any number and take the cube root of any perfect cube

⚡

Read the last digit of a cube root straight off the number

🔢

Find a cube root by grouping prime factors in threes

💧

Explain why doubling a tank in every direction gives eight times the water

“

A square number fills a flat square. A cube number fills a solid cube. Eight small blocks really do stack into a 2 by 2 by 2 cube.

🔐

8 more slides are waiting

Create a free account to watch the full presentation.

Create free account Sign in

Presenter notes: Open with blocks, exactly as the last lesson opened with counters. Hand each group a pile of sugar lumps or small wooden cubes and ask them to build the smallest solid cube they can that is bigger than a single block. They will build 2 by 2 by 2 and count 8. Ask for the next one up, and they will need 27. The jump from 8 to 27 is startling when you are holding the blocks, and that surprise is the whole subject of the lesson.