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

1 Answer
Jun 6, 2017

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.