Cubes and cube roots
Cubic numbers and cube roots often appear when working with volume – like figuring out how much water a cube-shaped tank can hold, or how big a box needs to be to store a certain number of smaller cubes. Understanding cubes and their roots helps you think in three dimensions and solve problems involving space, design, and engineering. Jump to the questions
Practise now
Solve the following cubes and cube roots.
Topic guide
What this worksheet practises
This worksheet provides practice on calculating cubes and cube roots. A cube number is the result of multiplying a number by itself, and then by itself again. The cube root is the exact opposite operation. These calculations are essential for finding the volume of 3D shapes.
Key method
Cubing a number and finding a cube root are inverse (opposite) operations.
- To cube a number (written as x³), calculate: x × x × x.
- To find a cube root (written as ³√x), you are looking for the number which, when multiplied by itself three times, gives 'x'.
- Memorising the first five cube numbers is highly recommended to save time in non-calculator exams.
Worked example
Calculate 4³ and then find the cube root of 125.
Step 1: Calculate 4³.
4³ means 4 × 4 × 4.
4 × 4 = 16.
16 × 4 = 64.
So, 4³ = 64.
Step 2: Find ³√125.
We need a number that, multiplied by itself three times, equals 125. Since it ends in a 5, the root is very likely 5.
5 × 5 × 5 = 25 × 5 = 125.
So, ³√125 = 5.
Common mistakes to avoid
The most frequent error is multiplying the base number by 3 instead of cubing it. For example, incorrectly thinking that 4³ = 4 × 3 = 12. You must multiply the number by itself. Another mistake is confusing square roots with cube roots when looking at the radical symbol; always look for the small '3' outside the root.
Things to remember
The first five cube numbers are: 1, 8, 27, 64, and 125. Knowing these off by heart will make exam questions significantly easier and faster to solve.