Question

How do you simplify `root3(3/4)`?

Answer 1

`root(3)(3/4) = root(3)(6)/2`

Explanation:

For any non-zero values of `a, b` we have:

`root(3)(a/b) = root(3)(a)/root(3)(b)`

`root(3)(a^3) = a`

So we find:

`root(3)(3/4) = root(3)((3*2)/(4*2)) = root(3)(6/2^3) = root(3)(6)/root(3)(2^3) = root(3)(6)/2`

Notice how making the denominator into a perfect cube before splitting the radical allows us to avoid having to rationalise the denominator afterwards.

Answer 2

`color(blue)(root3(6)/2`

Explanation:

`root3(3/4)`

`:.=root3(3)/root3(4) xx root3(4)/root3(4) xx root3(4)/root3(4)`

`:.=color(blue)(root3(4)*root3(4)*root3(4)=4`

`:.=root3(48)/4`

`:.=root3(3*2*2*2*2)/4`

`:.=(cancel2^color(blue)1root3(6))/cancel4^color(blue)2`

`:.=color(blue)(root3(6)/2`