MathematicsExam 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^(1/n) = ⁿ√a | A unit fraction index is a root | 16^(1/2) = 4 |
| a^(m/n) = (ⁿ√a)ᵐ | Root first, then power | 27^(2/3) = 3² = 9 |
- Taking the root before applying the power keeps the numbers small, which matters without a calculator
- A negative index never makes the answer negative: it makes it a reciprocal
Worked example
Negative and fractional powers by hand
Without a calculator, find the value of 64^(−2/3) and 81^(3/4).
Solution:
- Leave the minus sign until last: 64^(−2/3) = 1 ÷ 64^(2/3)
- The denominator of the index is 3, so take the cube root: ∛64 = 4
- The numerator is 2, so square the result: 4² = 16
- So 64^(2/3) = 16, and 64^(−2/3) = 1/16
- For 81^(3/4), the denominator 4 means a fourth root: 3⁴ = 81, so ⁴√81 = 3
- The numerator 3 means cube it: 3³ = 27
- Rooting first keeps the numbers small: cubing 81 first gives 531 441, whose fourth root is far harder to find by hand
Exam tip
Write down the root or the reciprocal step
When finding a fractional or negative power is worth two marks, one is for a written step and one for the exact answer. For a positive fractional power the root on its own scored (∛216 = 6 for 216^(2/3)); for a negative power the step that scored was the reciprocal, such as 1/3³ or 27⁻¹ for 9^(−3/2). Write that line before the final value: if the last step slips, it still earns a mark.
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 solve 4ˣ = 8.
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 equation, the two sides have different bases, so rewrite both as powers of 2
- 4 = 2², so 4ˣ = (2²)ˣ = 2²ˣ, and 8 = 2³
- The equation becomes 2²ˣ = 2³
- Equal powers of the same base must have equal indices, so 2x = 3 and x = 1.5