Question

How do you rationalize the denominator and simplify `5/(sqrt3-1)`?

Answer 1

`(5sqrt3+5)/2=(5/2)(sqrt3+1)`

Explanation:

We can simplify this fraction by using a clever use of the number 1 and also by keeping in mind that something in the form of:

`(a+b)(a-b)=a^2-b^2`, so that will allow us to rationalize the denominator without getting into messy square root issues.

Starting with the original:

`5/(sqrt3-1)`

We can now multiply by 1 and not change the value of the fraction:

`5/(sqrt3-1)(1)`

and we can now pick a value of 1 that will suit our purposes. We have something in the form of `(a-b)` already in the denominator, so let's multiply by `(a+b)`:

`5/(sqrt3-1)((sqrt3+1)/(sqrt3+1))`

which gets us:

`(5(sqrt3+1))/((sqrt3-1)(sqrt3+1))`

simplifying:

`(5sqrt3+5)/(3-1)=(5sqrt3+5)/2=(5/2)(sqrt3+1)`