Question

How do you simplify `1/{1+sqrt(3)-sqrt(5)}`?

Answer 1

`(7 + 3sqrt(3) + sqrt(5) + 2sqrt(15))/(11)`

Explanation:

You're going to have to do a little work here to simplify this expression.

The way to go is by rationalizing the denominator. The only problem is the fact that your denominator is a trinomial, and conjugates are only formed for binomials.

More specifically, you get the conjugate of a binomial by changing the sign of its second term.

`a + b -> underbrace(a color(red)(-) b)_(color(blue)("conjugate"))" "` or `" "a - b -> overbrace(a color(red)(+) b)^(color(blue)("conjugate"))" "`

This means that you're going to have to group the denominator as a binomial, for which you can write

`overbrace(1)^(color(red)(a)) + overbrace((sqrt(3) - sqrt(5)))^(color(red)(b)) -> underbrace(1 color(red)(-) (sqrt(3) - sqrt(5)))_(color(blue)("conjugate"))`

So, multiply your expression by `1 = (1 - (sqrt(3) - sqrt(5)))/(1 - (sqrt(3) - sqrt(5)))` to get

`1/(1 + (sqrt(3) - sqrt(5))) * (1 - (sqrt(3) - sqrt(5)))/(1 - (sqrt(3) - sqrt(5)))`

`(1 - sqrt(3) + sqrt(5))/([1 + (sqrt(3) - sqrt(5))][1 - (sqrt(3) - sqrt(5))]`

The denominator can be rewritten as

`[1 + (sqrt(3) - sqrt(5))][1 - (sqrt(3) - sqrt(5))] = 1^2 - (sqrt(3) - sqrt(5))^2`

This, in turn, will be equal to

`1 - ((sqrt(3))^2 - 2sqrt(3 * 5) + (sqrt(5))^2) = 1 - 3 + 2sqrt(15) - 5`

`=2sqrt(15) - 7`

The expression becomes

`(1 - sqrt(3) + sqrt(5))/(2sqrt(15) - 7)`

Now do the same thing with the new denominator, i.e. find its conjugate

`2sqrt(15) - 7 -> 2sqrt(15) color(red)(+) 7`

and multiply the expression by `1 = (2sqrt(15) + 7)/(2sqrt(15) + 7)` to get

`(1 - sqrt(3) + sqrt(5))/(2sqrt(15) - 7) * (2sqrt(15) + 7)/(2sqrt(15) + 7)`

`((1- sqrt(3) + sqrt(5))(2sqrt(15) + 7))/((2sqrt(15) - 7)(2sqrt(15) + 7))`

The denominator will be equal to

`(2sqrt(15) - 7)(2sqrt(15) + 7) = (2sqrt(15))^2 - 7^2`

`=4 * 15 - 49 = 11`

The numerator will be

`(1 - sqrt(3) + sqrt(5))(2sqrt(15) + 7)`

`2sqrt(15) - 2sqrt(45) + 2sqrt(75) + 7 - 7sqrt(3) + 7sqrt(5)`

`2sqrt(15) - 6sqrt(5) + 10sqrt(3) + 7 - 7sqrt(3) + 7sqrt(5)`

`7 + 3sqrt(3) + sqrt(5) + 2sqrt(15)`

The simplified expression will thus be

`1/(1 + sqrt(3) - sqrt(5)) = color(green)( (7 + 3sqrt(3) + sqrt(5) + 2sqrt(15))/(11))`