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

1 Answer
Jun 6, 2017

Answer:

#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.