Question

How do you evaluate `\frac { \sqrt { 5} + 1} { 4+ \sqrt { 2} }`?

Answer 1

`(4-sqrt2+4sqrt5-sqrt10)/14`

Explanation:

The thing we want to be rid of is the square root in the denominator. We can do that by multiplying by a term to get rid of it:

`(sqrt5+1)/(4+sqrt2)(1)`

`(sqrt5+1)/(4+sqrt2)((4-sqrt2)/(4-sqrt2))`

And now we use FOIL for numerator and denominator:

`(4sqrt5-sqrt10+4-sqrt2)/(16-4sqrt2+4sqrt2-2)`

`(4-sqrt2+4sqrt5-sqrt10)/14`