Calculating with Fractional Indices
# Calculating with Fractional Indices

GCSE(H),

For a fractional index, the numerator is a power term, and the denominator is a root term.

`27^frac(2)(3)` means square 27 then cube root the answer; or cube root 27 then square the answer. Choose the easiest calculation. In this instance, the cube root of 27 is 3, and 3 squared is 9.

## Examples

1. Simplify `sqrt(2) xx root(4)2`, and show with a fractional index.

Answer: `2^frac(3)(4)`

`sqrt(2) xx root(4)2`

= `2^frac(1)(2) xx 2^frac(1)(4)`

add indices (Laws of Indices)

= `2^frac(3)(4)`

2. Simplify √5 ÷ 25. Leave the index as a fraction.

Answer: `5^(frac(-3)(2)`

`sqrt(5) ÷ 25`

`5^frac(1)(2) ÷ 5^2`

`5^(frac(1)(2) - 2)`

`5^-frac(3)(2)`

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