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?

Guest Nov 10, 2022

#1**+1 **

If we write \(\sqrt2 + \sqrt3 + \frac{1}{(2\sqrt2+ 3\sqrt3)}\) in the form \(\frac{(a\sqrt2 + b\sqrt3)}{c}\) such that a, b, and c are positive integers and c is as small as possible, then what is a + b + c?

\(\sqrt2 + \sqrt3 + \frac{1}{(2\sqrt2+ 3\sqrt3)}\\ =\sqrt2 + \sqrt3 + \frac{1}{(2\sqrt2+ 3\sqrt3)}*\frac{(2\sqrt2- 3\sqrt3)}{(2\sqrt2- 3\sqrt3)}\\ =\sqrt2 + \sqrt3 + \frac{(2\sqrt2- 3\sqrt3)}{(8- 27)}\\\)

You can finish it.

Melody Nov 10, 2022