Powers, roots and index laws

A power (or index) tells you how many times to multiply a number by itself: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Squaring is a power of 2, and cubing is a power of 3.

A root undoes a power: √36 = 6 because 6² = 36, and ∛27 = 3 because 3³ = 27.

1. Work out powers and roots before × ÷ + − (BIDMAS).
2. Put a negative number in brackets before you square it.

Example: work out 2⁵ + √81 − 4³.

1. 2⁵ = 32
2. √81 = 9
3. 4³ = 64
4. 32 + 9 − 64 = −23

These laws only work when the base (the number or letter being raised to a power) is the same. You must learn them.

With numbers in front, deal with the numbers and the letters separately.

Example: simplify 3a²b × 4a⁵b³.

1. Numbers: 3 × 4 = 12
2. a: a² × a⁵ = a⁷
3. b: b¹ × b³ = b⁴
4. 12a⁷b⁴

Example: simplify (2x³)².

1. Square everything in the bracket: 2² × (x³)²
2. 4 × x⁶ = 4x⁶

A negative power means the reciprocal (1 over the number):

x⁻¹ = 1/x, x⁻² = 1/x², and so on. Learn this rule.

1. Work out the positive power.
2. Take the reciprocal: put 1 over it, or flip a fraction.

Example: work out 5⁻² and 2⁻³.

1. 5² = 25, so 5⁻² = 1/25
2. 2³ = 8, so 2⁻³ = 1/8

Harder example: work out (2/3)⁻².

1. Flip the fraction: (3/2)²
2. 3² = 9 and 2² = 4
3. 9/4 (or 2 ¼)

The index laws still work: x³ ÷ x⁵ = x⁻², which is 1/x².

A fractional power means a root: power ½ is the square root (25 to the power ½ = 5), and power ⅓ is the cube root (8 to the power ⅓ = 2).

1. Take the root given by the denominator.
2. Raise it to the power given by the numerator.
3. If the power is negative, take the reciprocal.

Example: work out 27 to the power ⅔.

1. Cube root: ∛27 = 3
2. Square it: 3² = 9

Harder example: work out 16 to the power −¾.

1. Fourth root: the fourth root of 16 is 2, because 2⁴ = 16
2. Cube it: 2³ = 8
3. Negative power, so take the reciprocal: 1/8