Maths · Number & Negatives
Squares & cubes
Squares, cubes, and their roots.
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What are squares and cubes?
A square number is made when you multiply a whole number by itself. For example, 4 × 4 = 16, so 16 is a square number. We write this as 4². A cube number is made when you multiply a number by itself three times. For example, 3 × 3 × 3 = 27, so 27 is a cube number. We write this as 3³.
Square roots and cube roots
A square root reverses squaring. The square root of 16 is 4 because 4² = 16. We write this as √16 = 4. A cube root reverses cubing. The cube root of 27 is 3 because 3³ = 27. We write this as ∛27 = 3. Finding roots means asking what number was multiplied to make the result.
Special patterns to notice
The first few square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. The first few cube numbers are 1, 8, 27, 64, 125. Notice that 64 appears in both lists! Also, 1² = 1 and 1³ = 1, which makes 1 very special.
Working with negatives
When you square a negative number, the answer is positive. For example, (−3)² = (−3) × (−3) = 9. But when you cube a negative number, the answer stays negative. For example, (−2)³ = (−2) × (−2) × (−2) = −8. Remember that two negatives make a positive, but three negatives make a negative.
Worked examples
1. Calculate 5² + 2³
- First work out 5²: that means 5 × 5 = 25.
- Next work out 2³: that means 2 × 2 × 2 = 8.
- Now add them together: 25 + 8 = 33.
2. Find the value of √49 − ∛8
- √49 asks what number times itself equals 49. That's 7 because 7 × 7 = 49.
- ∛8 asks what number cubed equals 8. That's 2 because 2 × 2 × 2 = 8.
- Now subtract: 7 − 2 = 5.
3. Work out (−4)² + (−1)³
- (−4)² means (−4) × (−4) = 16. Two negatives multiply to give a positive.
- (−1)³ means (−1) × (−1) × (−1) = −1. Three negatives multiply to give a negative.
- Add them: 16 + (−1) = 15.
Try it yourself
Calculate 6² − 3³
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