Algebra

Find positive integers (a,b) so that sqrt(37 + 12*sqrt(7)) = a + b*sqrt(7).

$\begin{array}{|rcll|} \hline \sqrt{37 + 12*\sqrt{7}} &=& \sqrt{9+28 + 12*\sqrt{7}} \\ &=& \sqrt{9+ 12*\sqrt{7} + 28 } \\ &=& \sqrt{9+ 2*6*\sqrt{7} + 28 } \\ &=& \sqrt{\left(3+2\sqrt{7}\right)^2 } \\ \mathbf{\sqrt{37 + 12*\sqrt{7}} } &=& \mathbf{3+2\sqrt{7} } \\ \hline \end{array}$