Question

How do you rationalize the denominator and simplify `1/(sqrt2+sqrt5+sqrt3)`?

Answer 1

Refer to explanation

Explanation:

We multiply and divide with `sqrt2+sqrt3-sqrt5` hence

`(sqrt2+sqrt3-sqrt5)/((sqrt2+sqrt3+sqrt5)*(sqrt2+sqrt3-sqrt5))= (sqrt2+sqrt3-sqrt5)/((sqrt2+sqrt3)^2-(sqrt5)^2)= (sqrt2+sqrt3-sqrt5)/(2+3+2sqrt2sqrt3-5)= (sqrt2+sqrt3-sqrt5)/(2sqrt2sqrt3)= (sqrt2+sqrt3-sqrt5)/(2sqrt6)= (sqrt6*(sqrt2+sqrt3-sqrt5))/(2sqrt6*sqrt6)= (sqrt12+sqrt18-sqrt30)/12`

Finally we have

`1/((sqrt2+sqrt3+sqrt5))=(sqrt12+sqrt18-sqrt30)/12= (2sqrt3+3sqrt2-sqrt30)/12`