Integers
Square roots and cube roots
Squaring and square-rooting undo each other. Knowing the square numbers by heart makes the rest almost automatic.
Square numbers
2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, 5 × 5 = 25. The square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, … You write them with an INDEX of 2: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25. Read 2² as 'two squared'.
Square roots
4² = 16 is equivalent to 4 = √16, read as 'four is the square root of sixteen'. The symbol for square root is √. Worked example: √100 − √81. 10² = 100, so √100 = 10. 9² = 81, so √81 = 9. So √100 − √81 = 10 − 9 = 1.
Cube numbers and cube roots
1 × 1 × 1 = 1, 2 × 2 × 2 = 8, 3 × 3 × 3 = 27. The cube numbers are 1, 8, 27, 64, 125, … You use an index of 3: 1³ = 1, 2³ = 8, 3³ = 27. Read 2³ as 'two cubed'. 2 is the cube root of 8, written ∛8 = 2. Worked example: √64 ÷ ∛64. Since 8² = 64, √64 = 8. Since 4³ = 64, ∛64 = 4. So √64 ÷ ∛64 = 8 ÷ 4 = 2.
Estimating roots
Most numbers are not square numbers, but you can still trap their roots between two whole numbers. Is 9 the closest integer to √79? 9² = 81 and 8² = 64. Since 79 lies between 64 and 81, √79 lies between 8 and 9. And 79 is much closer to 81 than to 64 — so 9 is the closest integer. Show that √215 lies between 14 and 15: 14² = 196 and 15² = 225, and 215 sits between them.
Worked examples
√100 − √81
- 10² = 100, so √100 = 10.
- 9² = 81, so √81 = 9.
- 10 − 9 = 1.
Answer: 1
√64 ÷ ∛64
- 8² = 64, so √64 = 8.
- 4³ = 64, so ∛64 = 4.
- 8 ÷ 4 = 2.
Answer: 2
Is 9 the closest integer to √79?
- 8² = 64 and 9² = 81.
- 79 lies between them, so √79 is between 8 and 9.
- 79 is much nearer 81 than 64.
Answer: Yes, 9
Show that √215 lies between 14 and 15
- 14² = 196.
- 15² = 225.
- 215 sits between 196 and 225.
Answer: So √215 is between 14 and 15
Working out 5²
- The index 2 means 'multiply by itself'.
- 5 × 5 = 25.
Answer: 25
Working out 3³
- The index 3 means multiply it by itself three times.
- 3 × 3 × 3 = 27.
Answer: 27
Where you see this in real life
- A square patio of 49 tiles must be 7 tiles along each side, because √49 = 7.
- A cube-shaped box holding 64 unit cubes is 4 across, 4 deep and 4 high, because ∛64 = 4.
Key words
- Square number
- the result of multiplying a whole number by itself
- Square root
- the number that was squared, written √
- Cube number
- the result of multiplying a whole number by itself twice
- Cube root
- the number that was cubed, written ∛
- Index
- the small raised number showing a power
- Consecutive
- following one after another
- Squared
- multiplied by itself, shown by an index of 2
- Cubed
- multiplied by itself three times, shown by an index of 3