Question

What is `root3(32)/(root3(36))`? How do you rationalize the denominator, if needed?

Answer 1

I got: `2root3(81)/9`

Explanation:

Let us write it as:
`root3(32/36)=root3((cancel(4)*8)/(cancel(4)*9))=root3(8)/root3(9)=2/root3(9)`
rationalize:
`=2/root3(9)*root3(9)/root3(9)*root3(9)/root3(9)=2root3(81)/9`

Answer 2

or `(2root3(3))/3`

Explanation:

Given `root 3 (32)/root 3(36)` for rationalizing of denominator if required.
`root 3(32/36)`
Dividing the numerator and denominator by common factor 4.
or `root 3(cancel32^8/cancel36_9)`
or `root 3(8/9)`

or `2/root 3((3^2)`

[Since `8=2^3`, numerator 8 can be written as `root 3(2^3)=2`.
And denominator 9 can be written as `root 3(3^2)`].

We see that in order to make the exponent of the denominator equal to nearest whole number 1, we need to multiply it by `root 3(3)`.

Therefore, multiplying and dividing the numerator and denominator with `root 3(3)`
or `2*1/root3(3^2)*root 3(3)/root 3(3)`

or `2*root3(3)/3`