Question

How do you simplify `(- 216x^{3} y^{9} )^{\frac{1}{3}}`?

Answer 1

`-6xy^3`

Explanation:

Compare a generic case of `(-a)^3 = -a^3`

So the reverse process shows that the cube root of a negative number is also negative.

Note that `(y^9)^(1/3) -> y^(9/3) = y^3` and that `(x^3)^(1/3) ->x^(3/3)=x`

Now we can write this as `" "color(blue)(-xy^3(216)^(1/3))`

All we need to do now is find the cube root of 216

If you can not spot the values draw a quick sketch of a prime factor tree.

From this we observe that `216 = 2^3xx3^3` giving

`-xy^3(2^3xx3^3)^(1/3)" " =" "- xy^3xx2xx3" " =" " -6xy`