Question

Simplify `rootx(a^y)`? for example, `root8(5^6)`

Answer 1

`5^{\frac{3}{4}}`

Explanation:

`root(x)a` can also be represented as `a^{\frac{1}{x}}`, for instance `sqrt4 =4^(1/2)`. Therefore `root(x){a^y} = (a^y)^{\frac{1}{x}}`, and with indices laws we can simplify that to `a^{y*\frac{1}{x}} = a^{\frac{y}{x}}`.
In the case of `root(8){5^6} = 5^{\frac{6}{8}}`, which simplifies to `5^{\frac{3}{4}}`, which you could write as `root(4){5^3}`.
Apart from that however you can't simplify whilst keeping it in exact form.