Mathematics (Core)Exam code: 0580
Powers, Roots and Standard Form
Number
The rules
- The multiplying and dividing laws apply only where the terms share the same base
| Law | What it does | Example |
|---|---|---|
| aᵐ × aⁿ = aᵐ⁺ⁿ | Multiplying adds the indices | 5⁶ × 5³ = 5⁹ |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | Dividing subtracts the indices | 9⁷ ÷ 9⁴ = 9³ |
| (aᵐ)ⁿ = aᵐⁿ | A power of a power multiplies the indices | (2⁵)³ = 2¹⁵ |
| (ab)ⁿ = aⁿbⁿ | A power spreads across a product | (4 × 7)² = 4² × 7² |
| a⁰ = 1 | Any non-zero base to the power 0 | 12⁰ = 1 |
| a⁻ⁿ = 1 ÷ aⁿ | A negative index gives the reciprocal | 3⁻² = 1/9 |
- A negative index never makes the answer negative: it makes it a reciprocal
Worked example
Roots, zero and negative powers, without a calculator
Find the value of (a) ∛64, (b) 9⁰ and (c) 2⁻³, giving (c) as a fraction and as a decimal.
Solution:
- (a) 4³ = 64, so ∛64 = 4
- (b) Any non-zero number to the power 0 is 1, so 9⁰ = 1
- (c) A negative index means the reciprocal: 2⁻³ = 1 ÷ 2³ = 1/8
- As a decimal, 1 ÷ 8 = 0.125
Making the bases match
- Where the two bases differ but one is a power of the other, rewrite so that they match, and the laws then apply
- 4 = 2², so 2⁵ × 4³ = 2⁵ × (2²)³ = 2⁵ × 2⁶ = 2¹¹
- Powers of 2, 3, 5 and 7 are the ones worth recognising, since those are the bases questions use most
- Bases with no such relationship stay apart, so 2³ × 5² cannot be shortened by these laws at all
Worked example
Using the laws, and matching the bases
Simplify 2⁵ × 2³ ÷ 2², and write 25 × 125 as a single power of 5.
Solution:
- Multiplying powers of the same base adds the indices: 2⁵ × 2³ = 2⁸
- Dividing subtracts them: 2⁸ ÷ 2² = 2⁶
- So the expression is 2⁶, which is 64
- For the second, write each number as a power of 5: 25 = 5² and 125 = 5³
- The bases now match, so add the indices: 25 × 125 = 5² × 5³ = 5⁵