Simplify/Factor: (2sqrt(12))/(sqrt(2)+sqrt(3)-sqrt(5))?

Its the denominator that you want to "rationalize", right? With three square roots, it takes a couple of steps. I'll ignore the numerator until the end:

First multiply by (√2 + √3 + √5)/(√2 + √3 + √5):

1/(√2 + √3 - √5) = (√2 + √3 + √5)/[(√2 + √3 - √5)(√2 + √3 + √5)]

The denominator is a product of a sum and difference, which becomes a difference of squares:

= (√2 + √3 + √5) / [(√2 + √3)² - 5]

= (√2 + √3 + √5) / [(2 + 2√2√3 + 3) - 5]

= (√2 + √3 + √5) / (2√6)

The non-radical terms canceled out (this *never* happens for me...and I made an error here on the first version) so now multiply by √6/√6 to rationalize that denominator:

= (√12 + √18 + √30) / 12 .... remove square factors

= (2√3 + 3√2 + √30) / 12

Finally multiply in the original numerator 2√12 (which is 4√3)

2√12 / (√2 + √3 - √5) = (4√3)(2√3 + 3√2 + √30) / 12

= (24 + 12√6 + 4√90) / 12 .... then remove square factors

= (24 + 12√6 + 12√10) / 12 .... and a common factor of 12 cancels

= 2 + √6 + √10

The only simplification that I can see is in the numerator.

Split the sqrt(12):

2sqrt(12) = 2 * sqrt(4) * sqrt(3)

Take the square root of 4 and multiply:

2 * sqrt(4) * sqrt(3) = 2 * 2 * sqrt(3) = 4sqrt(3)