Question

How do you rationalize the denominator and simplify `4/(6-sqrt5)`?

Answer 1

Multiply both numerator and denominator by `(6+sqrt(5))` and simplify to find:

`4/(6-sqrt(5)) =(24+4sqrt(5))/31`

Explanation:

`4/(6-sqrt(5)) = (4(6+sqrt(5)))/((6-sqrt(5))(6+sqrt(5))) =`

`=(24+4sqrt(5))/(6^2-sqrt(5)^2) = (24+4sqrt(5))/(36-5)`

`=(24+4sqrt(5))/31`

This uses the difference of squares identity to square any square roots in the binomial denominator.

`(a-b)(a+b) = a^2 - b^2`

In our case `a = 6` and `b = sqrt(5)`, but it would work if both `a` and `b` were square roots.