Roots are operations that reverse powers. The square root of a number is the value which, when multiplied by itself, gives the original number. The cube root of a number is the value which, when multiplied by itself three times, gives the original number.
Square roots
The symbol for square root is √. If a × a = b, then √b = a. For example, since 4 × 4 = 16, √16 = 4.
| Number | Square root | Reason |
|---|---|---|
| 1 | 1 | 1 × 1 = 1 |
| 4 | 2 | 2 × 2 = 4 |
| 9 | 3 | 3 × 3 = 9 |
| 16 | 4 | 4 × 4 = 16 |
| 25 | 5 | 5 × 5 = 25 |
| 36 | 6 | 6 × 6 = 36 |
Perfect squares are numbers that have whole-number square roots. Examples include 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.
Finding square roots by prime factorisation
To find the square root of a larger perfect square, first express the number as a product of prime factors. Group equal factors in pairs, then take one factor from each pair.
For example: √144 = √(2 × 2 × 2 × 2 × 3 × 3) = 2 × 2 × 3 = 12.
For a fraction, find the square root of the numerator and denominator separately when both are perfect squares: √(9/16) = √9 / √16 = 3/4.
Cube roots
The symbol for cube root is ∛. If a × a × a = b, then ∛b = a. For example, since 2 × 2 × 2 = 8, ∛8 = 2.
| Number | Cube root | Reason |
|---|---|---|
| 1 | 1 | 1 × 1 × 1 = 1 |
| 8 | 2 | 2 × 2 × 2 = 8 |
| 27 | 3 | 3 × 3 × 3 = 27 |
| 64 | 4 | 4 × 4 × 4 = 64 |
| 125 | 5 | 5 × 5 × 5 = 125 |
To find a cube root by prime factorisation, group equal prime factors in threes and take one factor from each group.
Elementary sequences
An elementary sequence is a list of numbers written in a definite order. Each number in the list is called a term. The rule explains how one term changes into the next term.
| Sequence | Rule | Next terms |
|---|---|---|
| 2, 4, 6, 8, … | Add 2 | 10, 12, 14 |
| 1, 3, 5, 7, … | Add 2 | 9, 11, 13 |
| 3, 6, 9, 12, … | Add 3 | 15, 18, 21 |
| 2, 6, 18, … | Multiply by 3 | 54, 162, 486 |
To continue a sequence, compare consecutive terms and identify the operation used. The operation may be addition, subtraction, multiplication or division.
Worked examples
- Continue 1, 3, 5, 7, … by adding 2: the next terms are 9, 11 and 13.
- Continue 62, 55, 48, … by subtracting 7: the next terms are 41, 34 and 27.
- Continue 5, 9, 13, … by adding 4: the next terms are 17, 21 and 25.
Summary
- A square root reverses a square: √a is the number which gives a when multiplied by itself.
- A cube root reverses a cube: ∛a is the number which gives a when multiplied by itself three times.
- Prime factors are grouped in pairs for square roots and in groups of three for cube roots.
- A sequence is an ordered list of numbers governed by a rule.
- Each member of a sequence is called a term.
Practice questions
- Find √36, √49, √64, √81 and √100.
- Find ∛8, ∛27, ∛64 and ∛125.
- Find √(25/36).
- Write the next three terms: 2, 6, 10, …; 7, 9, 11, …; and 2, 6, 18, ….
