Question

If a=8+3√2, b=1/a. Find a^2+b^2

Answer 1

${ \boxed{ \huge{a = 8 + 3 \sqrt{2}}}} \\ \\ { \boxed{ \huge{b = \frac{1}{8 + 3 \sqrt{2} } = \frac{8 - 3 \sqrt{2} }{64 - 18} = { \red{\frac{8 - 3 \sqrt{2} }{46} }} }}}$

Now,

${ \large{{a}^{2} + {b}^{2} = {(a + b)}^{2} - 2ab }} \\ \\ = > { \boxed{ \large{ (8 + 3 \sqrt{2} + \frac{8 - 3 \sqrt{2} }{46} )^{2} - 2(8 + 3 \sqrt{2} )( \frac{8 - 3 \sqrt{2} }{46} )}}} \\ \\ \\ = > { \boxed{ \large{ {(8 + 3 \sqrt{2} )}^{2} + 2(8 + 3 \sqrt{2} )( \frac{8 - 3 \sqrt{2} }{46} ) + (\frac{8 - 3 \sqrt{2} }{46})^{2} - 2(8 + 3 \sqrt{2} )( \frac{8 - 3 \sqrt{2} }{46} ) }}} \\ \\ \\ = > { \boxed{ \large{(64 + 18 + 48 \sqrt{2} ) + ( \frac{64 + 18 - 48 \sqrt{2} }{2116} )}}} \\ \\ \\ = > { \boxed{ \large{ \frac{135424 + 38088 + 101568 \sqrt{2} + 64 + 48 - 48 \sqrt{2} }{2116} }}} \\ \\ \\ = > { \boxed{ \large{ \frac{173512 + 101568 \sqrt{2} + 112 - 48 \sqrt{2} }{2116} }}} \\ \\ = > { \boxed{ \large{ { \red{\frac{173624 + 101520 \sqrt{2} }{2116}}} }}}$