+415 Compute the sum
\frac{1}{\sqrt{36} + \sqrt{39}} + \frac{1}{\sqrt{39} + \sqrt{40}} + \frac{1}{\sqrt{40} +\sqrt{45}}
+129771 1 1 1
_____________ + _____________ + _______________
sqrt 36 +sqrt 39 sqrt 39 + sqrt 40 sqrt 40 + sqrt 45
Multiplying numerartors/denominators by the conjugate of the denominators, we can write
sqrt 36 - sqrt 39 sqrt 39 -sqrt 40 sqrt 40 -sqrt 45
_______________ + ______________ + ______________
36 - 39 39 - 40 40 - 45
sqrt 39 - sqrt 36 sqrt 40 - sqrt 39 sqrt 45 - sqrt 40
_____________ + _______________ + ______________
3 1 5
(-2/3)sqrt 39 - 2 + (4/5)sqrt 40 + (1/5)sqrt 45
(-2/3)sqrt 39 - 2 + (4/5)* 2 sqrt 10 + (1/5) 3sqrt 5
(-2/3)sqrt 39 - 2 + (8/5)sqrt 10 + (3/5)sqrt 5
