If we write sqrt(2) + sqrt(3) + 1/(2*sqrt(2) + 3*sqrt(3)) in the form (a*sqrt(2) + b*sqrt(3))/c such that a, b, and c are positive integers and c is as small as possible, then what is a + b + c?
+23252 Rationalize the denominator of the term: 1 / [ 2·sqrt(2) + 3·sqrt(3) ]
by multiplying both the numerator and denominator by the conjugate of the
denominator which is: 2·sqrt(2) - 3·sqrt(3).
1 / [ 2·sqrt(2) + 3·sqrt(3) ] · [ 2·sqrt(2) - 3·sqrt(3) ] / [ 2·sqrt(2) - 3·sqrt(3) ]
= [ 2·sqrt(2) - 3·sqrt(3) ] / -19 = [ 3·sqrt(3) - 2·sqrt(2) ] / 19
Rewrite this as two separate fractions: 3·sqrt(3) / 19 - 2·sqrt(2) / 19
Now, write the other two terms with a donominator of 19:
sqrt(2) = 19·sqrt(2) / 19 and sqrt(3) = 19·sqrt(3) / 19
Then, simplify this: 19·sqrt(2) / 19 + 19·sqrt(3) / 19 + 3·sqrt(3) / 19 - 2·sqrt(2) / 19
= 17·sqrt(2) / 19 + 22·sqrt(3) / 19
= [ 17·sqrt(2) / 19 + 22·sqrt(3) ] / 19
From this, you can find the answer.
