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 First: write 1 / [ 2sqrt(2) + 3sqrt(3) ] with a rational denominator by multiplying both
the numerator and the denominator by the conjugate of the denominator:
2sqrt(2) - 3sqrt(3):
1 / [ 2sqrt(2) + 3sqrt(3) ] · [ 2sqrt(2) - 3sqrt(3) ] / [ 2sqrt(2) - 3sqrt(3) ]
= [ 2sqrt(2) - 3sqrt(3) ] / [ 8 - 27 ]
= [ 2sqrt(2) - 3sqrt(3) ] / - 19
= [ 3sqrt(3) - 2sqrt(2) / 19
Second: wrtie sqrt(2) + sqrt(3) with a denominator of 19:
= 19sqrt(2) / 19 + 19sqrt(3) / 19
Add together these two results: [ 3sqrt(3) - 2sqrt(2) / 19 + 19sqrt(2) / 19 + 19sqrt(3) / 19
= [ 22sqrt(3) + 17sqrt(2) ] / 19
a = 17 b = 22 c = 19
