Question

How do you evaluate `\frac { 4+ 2\sqrt { 5} } { 7+ 3\sqrt { 5} }`?

Answer 1

`frac(sqrt(5)-1)(2)`

Explanation:

Multiply both the numerator and the denominator by the conjugate of the denominator - switch the plus to a minus.
`frac(4+2sqrt(5))(7+3sqrt(5))*frac(7-3sqrt(5))(7-3sqrt(5))`

Multiply both the numerator and the denominator together.
`frac((4+2sqrt(5))(7-3sqrt(5)))((7+3sqrt(5))(7-3sqrt(5)))=frac(28-12sqrt(5)+14sqrt(5)-(6*5))(49-21sqrt(5)+21sqrt(5)-(9*5)`

Multiply inside the parentheses.
`frac(28-12sqrt(5)+14sqrt(5)-30)(49-21sqrt(5)+21sqrt(5)-45`

Combine like terms.
`frac(2sqrt(5)-2)(4)`

Simplify `2` from the numerator and the denominator.

`frac(sqrt(5)-1)(2)`