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\(Find positive integers (a,b) so that \sqrt{37 + 20 \sqrt{3}} = a + b \sqrt{3}. Enter your answer in the form "a, b".\)

 Apr 16, 2019

\(\sqrt{37+20\sqrt{3}}=a+b\sqrt{3}\)- 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 \(5,2\) .

 Apr 16, 2019

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