Expressions and Equations
Cubes and cube roots
Cube a number, recognize a perfect cube, and find a cube root by prime factorisation.
Cubing a number means using it three times in a multiplication: 4 cubed is 4 × 4 × 4 = 64. It is written 4³, and the name comes from the shape — a cube with edges of 4 holds exactly 64 unit cubes. Going the other way, the cube root of 64 is 4, because 4 is the number that was cubed.
The first ten cubes
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. Cubes grow far faster than squares, so this short list covers most of what you will be asked to recognize. Unlike squares, a cube keeps the sign of the number it came from: (-3)³ = -27.
Worked example
Find the cube root of 1728 by prime factorisation.
Break 1728 into primes: 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3.
Keep dividing by the smallest prime that goes in, until only primes are left.
Try it together
Now let us find the cube root of 2744 together.
Same method: primes first, then groups of three, then one from each group.
1.2744 breaks into 2 × 2 × 2 × 7 × 7 × 7. How many complete groups of three the same can you make?
Have a go
Have a go on your own: 9261 breaks into 3 × 3 × 3 × 7 × 7 × 7. What is its cube root?
Hint: One factor from each group of three, multiplied together.
Ready to practice?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.