0580
Algebraic Roots and Indices
Algebra and Sequences
Working with the same base
- The index rules apply to algebra exactly as they do to numbers, provided the terms 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
Worked example
Applying the index rules
Simplify x⁵ × x⁻² ÷ x⁴, and work out the value of 16^(3/4).
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
- For 16^(3/4), the denominator of the fraction is the root and the numerator is the power
- Take the fourth root first, because it keeps the numbers small: ⁴√16 = 2
- Then raise to the power 3: 2³ = 8
- Doing it the other way round means cubing 16 to 4096 first, which is the same answer by a much harder route