Question

How do you simplify `root3(x) / root 3(2)`?

Answer 1

`root3(x/2)`

Explanation:

`sqrt(a)/sqrt(b) = sqrt(a/b)`

Answer 2

`(root(3)(4x))/2`

Explanation:

Usually, you do not want a radical in the denominator. In other words, you usually want to rationalize the denominator.

Here, the fraction can be expressed as `x^(1/3)/2^(1/3)`. If you multiply this by `2^(2/3)/2^(2/3)` (which is equal to `1`), the denominator is rationalized. To see how, remember that `a^b*a^c=a^(b+c)`. Then, `x^(3/2)/2^(3/2)*2^(1/2)/2^(1/2)=(x^(1/3)*2^(2/3))/(2^(1/3)*2^(2/3))=(x^(1/3)*2^(2/3))/(2^(1/3+2/3))=(x^(1/3)*2^(2/3))/4=(x^(1/3)*2^(2/3))/2=(root(3)(x)*root(3)(4))/2=(root(3)(4x))/2`.