The question is "Find the exact value of $\sqrt{97+56\sqrt{3}}$ ". It's from some regional contest back in 2013 and the answer is $7+4\sqrt{3}$. Can someone explain how they can reach the answer other than bashing in perfect squares for 97? I know you put $97+56\sqrt{3}$ into the form $a^2+2ab+b^2$ but how?
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the main thing to check is $97^2 - 3 \cdot 56^2$ which turns out to be one. So you are looking for $x^2 - 3 y^2 = \pm 1$ – Will Jagy Feb 15 '24 at 03:07
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Thanks for all who commented! They're all helpful and now I know how to approach these problems! – Algi King Feb 15 '24 at 03:27