MathematicsExam code: 0580
Algebraic Roots and Indices
Algebra and Sequences
Negative indices
- A negative index means the reciprocal of the positive power
- p⁻³ = 1/p³
- The sign of the index says nothing about the sign of the value
Fractional indices
- A unit fraction index is a root, with the denominator giving which root
- a^(1/2) is the square root and a^(1/3) is the cube root
- A general fraction m/n combines both: take the nth root, then raise to the power m
- 8^(2/3) is the cube root of 8, which is 2, then squared, giving 4
- Rooting before powering keeps the numbers small, which matters without a calculator
- A negative fractional index combines the two ideas, giving one over the root
Worked example
Simplifying a power of a product
Simplify (16x⁴)^(1/2).
Solution:
- The power applies to both the number and the letter
- The square root of 16 is 4
- For x⁴, halve the index: x⁴ raised to the power 1/2 gives x²
- The simplified form is 4x²
Worked example
A negative fractional power of a bracket
Simplify (81x⁸)^(−3/4).
Solution:
- The power applies to the 81 and to the x⁸ separately
- For the number, take the fourth root first: ⁴√81 = 3
- Then cube it: 3³ = 27, and the negative sign makes it the reciprocal, 1/27
- For the letter, multiply the indices: 8 × (−3/4) = −6, giving x⁻⁶ = 1/x⁶
- Putting the two parts together gives 1/(27x⁶), which can also be written (1/27)x⁻⁶