\(Find positive integers (a,b) so that \sqrt{37 + 20 \sqrt{3}} = a + b \sqrt{3}. Enter your answer in the form "a, b".\)
+4622 \(\sqrt{37+20\sqrt{3}}=a+b\sqrt{3}\)- Original Equation; trying to solve for \(a,b\).
Square both sides, to get \({37+20\sqrt{3}}=a^2+2ab\sqrt{3}+3b^2.\) This is just the expansion of \((a+b)^2\) !
Now, we use a bit of matching, on trying to match the variables to the numbers. We see \(20\sqrt{3}\) and \(2ab\sqrt{3}\) , so we have \(2ab\sqrt{3}=20\sqrt{3}\) , and \(ab=10.\)
Also, we get \(a^2+3b^2=37\) by matching the variables to the numbers, again.
After trying a few times, we get \(a=5, b=2\) , and that sure works !
Check: \((5)(2)=10\) and \(5^2+3(2)^2=25+12=37.\)
Thus, the answer is \(5,2\) .
