Question

How do you rationalize the denominator and simplify `5/(root3(4))`?

Answer 1

`frac{5}{root3 4}=frac{5root3 2}{2}`

Explanation:

Multiply the fraction by `frac{4^(2/3)}{4^(2/3)}`:

`frac{5}{root3 4} color(blue)(xx frac{root3 4}{root3 4} xx frac{root3 4}{root3 4})`

`=frac{5*4^(2/3)}{4}`

`=frac{5root3 16}{4}`

`=frac{5root3(8*2)}{4}`

`=frac{5*2*root3 2}{4}`

`=frac{5root3 2}{2}`

Answer 2

`5/root(3)(4) = (5root(3)(2))/2`

Explanation:

Given:

`5/root(3)(4)`

First note that:

`root(3)(4) = (2^2)^(1/3) = 2^(2/3)`

So to make the denominator rational it will be sufficient to multiply it by `root(3)(2)`...

`5/root(3)(4) = (5root(3)(2))/(root(3)(4)root(3)(2))`

`color(white)(5/root(3)(4)) = (5root(3)(2))/root(3)(4*2)`

`color(white)(5/root(3)(4)) = (5root(3)(2))/root(3)(2^3)`

`color(white)(5/root(3)(4)) = (5root(3)(2))/2`