Simplify by rationalizing the denominator?

Question

square root sign 11 - squareroot sign 5 / squareroot sign 11 + squareroot sign 5

Answer 1

(sqrt 11 - sqrt 5) / (sqrt 11 + sqrt 5)

Multiply the top and bottom by (sqrt 11 - sqrt 5)

Top:
(sqrt 11 - sqrt 5)(sqrt 11 - sqrt 5)
= 11 - 2(sqrt 11)(sqrt 5) + 5
= 16 - 2(sqrt 55)

Bottom:
(sqrt 11 + sqrt 5)(sqrt 11 - sqrt 5)
= 11 - 5
= 6

So, now you have:
(16 - 2sqrt 55) / 6
= 2(6 - sqrt 55) / 2*3
= (6 - sqrt 55) / 3

Answer 2

(√11 - √5) / (√11 + √5)
= (√11 - √5).(√11 - √5) / (11 + √5).(11 - √5)
= (11 - 2.√55 + 5) / (11 - 5)
= (1/6).(16 - 2.√55)
= (1/3).(8 - √55)