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?
+1632 The original expression is:
\(\sqrt{2} + \sqrt{3} + {1\over2\sqrt{2} + 3\sqrt{3}}\)
Next we can simplify the fraction part of the expression by multiplying the denominator by the difference of squares;
The expression is now:
\(\sqrt{2}+\sqrt{3} + {3\sqrt{3}-2\sqrt{2}\over19}\)
Then we can multiply the first two terms of the expression by \(19\) to get:
\({19\sqrt{2}\over19} + {19\sqrt{3}\over19} + {3\sqrt{3}-2\sqrt{2}\over19}\)
Then adding up the numerators and simplifying we get:
\(17\sqrt{2} + 22\sqrt{3}\over19\)
Thus, \(a\), \(b\), and \(c\) are \(17\), \(22\), and \(19\) respectively.
Since we are looking for the sum of our three "variables", we add \(17+22+19\) to get:
\(58\), which is the answer.
:D
+364 \(\sqrt{2}+\sqrt{3}+\left(\frac{1}{2\sqrt{2}+3\sqrt{3}}\right)\)
\(\sqrt{2}+\sqrt{3}+\frac{1}{2\sqrt{2}+3\sqrt{3}}\)
\(\frac{\sqrt{2}\left(2\sqrt{2}+3\sqrt{3}\right)}{2\sqrt{2}+3\sqrt{3}}+\sqrt{3}+\frac{1}{2\sqrt{2}+3\sqrt{3}}\)
\(\frac{\sqrt{2}\left(2\sqrt{2}+3\sqrt{3}\right)}{2\sqrt{2}+3\sqrt{3}}+\frac{\sqrt{3}\left(2\sqrt{2}+3\sqrt{3}\right)}{2\sqrt{2}+3\sqrt{3}}+\frac{1}{2\sqrt{2}+3\sqrt{3}}\)
\(\frac{\sqrt{2}\left(2\sqrt{2}+3\sqrt{3}\right)+\sqrt{3}\left(2\sqrt{2}+3\sqrt{3}\right)+1}{2\sqrt{2}+3\sqrt{3}}\)
\(\frac{14+5\sqrt{6}}{2\sqrt{2}+3\sqrt{3}}\)
\(-\frac{-17\sqrt{2}-22\sqrt{3}}{19}\)
\(\frac{17\sqrt{2}+22\sqrt{3}}{19}\)
17+19+22=58, just the same as proyaop's answer!
