Mathematics (Core)Exam code: 0580
Algebraic Roots and Indices
Algebra and Sequences
The Rules of Indices
Working with the same base
- The index rules apply to algebra exactly as they do to numbers, and the multiplying and dividing rules need the terms to share a base
| Rule | What it does | Example |
|---|---|---|
| aᵐ × aⁿ = aᵐ⁺ⁿ | Multiplying adds the indices | x⁵ × x³ = x⁸ |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | Dividing subtracts the indices | y⁹ ÷ y⁴ = y⁵ |
| (aᵐ)ⁿ = aᵐⁿ | A power of a power multiplies them | (m³)⁴ = m¹² |
| (ab)ⁿ = aⁿbⁿ | The power reaches every factor | (2a³)⁴ = 16a¹² |
| a⁰ = 1 | Any non-zero base to the power zero | 7x⁰ = 7 |
- A coefficient is raised to the power along with the letter, which is what turns 2 into 16 in the example above
- Terms with different bases cannot be combined by these rules, so x³ × y² stays as it is
Exam tip
The number and each power score separately
A two-mark simplification such as 4p³ × 5p⁶ = 20p⁹ is marked part by part: 20 with a wrong power of p, or p⁹ with a wrong number, still earns one mark, as does one letter's power right in a two-letter answer. The marks go on the final answer, so once each letter appears just once, stop: changing a correct answer into a wrong one costs one of the two marks.
Worked example
Multiplying terms with numbers in front
Simplify 6m⁴n × 3m²n³.
Solution:
- Deal with the numbers and with each letter separately
- Multiply the numbers: 6 × 3 = 18. They are multiplied, not added
- Add the powers of m: m⁴ × m² = m⁶
- n on its own is n¹, so add the powers of n: n¹ × n³ = n⁴
- The answer is 18m⁶n⁴
Worked example
Applying the index rules
Simplify x⁵ × x⁻² ÷ x⁴.
Solution:
- Multiplying powers of the same base adds the indices: 5 + (−2) = 3, giving x³
- Dividing subtracts them: 3 − 4 = −1, giving x⁻¹
- A negative index means a reciprocal, so the answer is 1/x
Worked example
Finding a missing index
Find the value of the letter in each equation: (a) m⁶ × mⁿ = m¹¹, (b) 4⁷ ÷ 4ᵏ = 4¹², (c) (q⁴)ʸ = q²⁴, (d) 9ᵗ = 1.
Solution:
- Once the left-hand side is written as a single power of the same base, its index must equal the index on the right
- (a) Multiplying adds the indices: 6 + n = 11, so n = 5
- (b) Dividing subtracts them: 7 − k = 12, so k = −5
- (c) A power of a power multiplies them: 4y = 24, so y = 6
- (d) Any non-zero number to the power 0 is 1, so 9ᵗ = 9⁰ and t = 0