Question

How do you simplify `sqrt3/(sqrt6 -1) - sqrt3/(sqrt6 + 1)`?

Answer 1

`(2 sqrt(3)) / 5`

Explanation:

You need to find the least common denominator to be able to subtract the two fractions.

In your case, the least common denominator is `(sqrt(6) - 1)(sqrt(6) + 1)`, so you need to expand the first fraction with `(sqrt(6) + 1)` and the second one with `(sqrt(6) - 1)`.

`sqrt(3) / (sqrt(6) - 1) - sqrt(3) / (sqrt(6) + 1) = (sqrt(3)color(blue)((sqrt(6) + 1))) / ((sqrt(6) - 1)color(blue)((sqrt(6) + 1))) - (sqrt(3)color(green)((sqrt(6) - 1))) / ((sqrt(6) + 1)color(green)((sqrt(6) - 1)))`

`= (sqrt(3)(sqrt(6) + 1) - sqrt(3)(sqrt(6) - 1)) / ((sqrt(6) - 1)(sqrt(6) + 1))`

... use the formula `(a + b)(a - b) = a^2 - b^2` to simplify the denominator...

`= (sqrt(3) * sqrt(6) + sqrt(3) - sqrt(3) * sqrt(6) + sqrt(3)) / ((sqrt(6))^2 - 1^2)`

`= (cancel(sqrt(3) * sqrt(6)) + sqrt(3) - cancel(sqrt(3) * sqrt(6)) + sqrt(3)) / (6 - 1)`

`= (2 sqrt(3)) / 5`

`= 2/5 sqrt(3)`