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 (You can also cube root 27 then square the answer; the order does not matter). Choose the easiest calculation. In this instance, the cube root of 27 is 3, and 3 squared is 9.
Simplify `sqrt(2) xx root(4)2`, and show with a fractional index.
`sqrt(2) xx root(4)2`
= `2^frac(1)(2) xx 2^frac(1)(4)`
add indices (Laws of Indices)
= `2^frac(3)(4)`
Answer: `2^frac(3)(4)`
Simplify √5 ÷ 25. Leave the index as a fraction.
`sqrt(5) ÷ 25`
`5^frac(1)(2) ÷ 5^2`
`5^((frac(1)(2) - 2))`
`5^-frac(3)(2)`
Answer: `5^(frac(-3)(2)`
See also Growth and Decay