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Rationalize the denominator of $\displaystyle \frac{1}{\sqrt{5} - \sqrt{2}}$. With your answer in the form $\displaystyle \frac{\sqrt{A} + \sqrt{B}}{C}$, and the fraction in lowest terms, what is $A + B + C$?

 
 Sep 8, 2021
 #1
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To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator.

 

The conjugate of  sqrt(5) - sqrt(2)  is  sqrt(5) + sqrt(2).

 

 1/ [ sqrt(5) - sqrt(2) ]   ·   [ sqrt(5) + sqrt(2) ] / [ sqrt(5) + sqrt(2) ] 

 

=  [ sqrt(5) + sqrt(2) ] / { [ sqrt(5) - sqrt(2) ] · [ sqrt(5) + sqrt(2) ] }

 

=  [ sqrt(5) + sqrt(2) ] / [ 5 - 2 ]

 

=  [ sqrt(5) + sqrt(2) ] / 3

 

A is 5, B is 2 and C is 3 ...

 
 Sep 8, 2021

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