How do you rationalize the denominator 1/ (1+ sqrt3 - sqrt5)?

1 Answer
Alan P.
Apr 3, 2015

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

The rationalization of this denominator is a two-set process.
First rationalize the sqrt(5)
and then, after some simplification
rationalize the sqrt(3)

Let a=(1+sqrt(3))
So our initial stage is to rationalize the denominator of
1/(a-sqrt(5))

As usual to do this we multiply the numerator and the denominator by the conjugate of the denominator:
1/(a-sqrt(5)) * (a+sqrt(5))/(a+sqrt(5))

= (a+sqrt(5))/(a^2 - 5)

Substituting (1+sqrt(3)) back in to this expression in place of a.
we get
= (1+sqrt(3)+sqrt(5))/((1+sqrt(3))^2-5)

=(1+sqrt(3)+sqrt(5))/(1 + 2sqrt(3) +3 -5)

=(1+sqrt(3)+sqrt(5))/ (2sqrt(3)-1)

Rationalizing this denominator (using the conjugate of 2sqrt(3)-1
results in
((1+sqrt(3)+sqrt(5)))/((2sqrt(3)-1)) * ((2sqrt(3)+1))/((2sqrt(3)+1))

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

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