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

Question

Nathan Axel · Accepted Answer

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

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))#

Eva Abramson · Answer

undefined To simplify the expression 1/{1+sqrt(3)-sqrt(5)}, we can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is 1+sqrt(3)+sqrt(5).

By doing this, we get:
1/{1+sqrt(3)-sqrt(5)} * (1+sqrt(3)+sqrt(5))/(1+sqrt(3)+sqrt(5))

Simplifying the numerator, we have:
1 + sqrt(3) + sqrt(5)

Expanding the denominator, we have:
(1+sqrt(3)-sqrt(5))(1+sqrt(3)+sqrt(5)) = 1 + sqrt(3) + sqrt(5) + sqrt(3) + 3 - sqrt(15) - sqrt(5) - sqrt(15) + 5
= 10 - 2sqrt(15)

Therefore, the simplified expression is:
1/{1+sqrt(3)-sqrt(5)} = (1 + sqrt(3) + sqrt(5))/(10 - 2sqrt(15))