Question

If x=8+3√7,
${x}^{2} + \frac{1}{ {x}^{2} }$
Explain by step by step solution

Answer 1

given x = 8 + 3√7


So, 1/x =
$\frac{1}{8 + 3 \sqrt{7} }$
now rationalize it

=>>
$\frac{1}{8 + 3 \sqrt{7} } \\ \\ = \frac{8 - 3 \sqrt{7} }{(8 + 3 \sqrt{7})(8 - 3 \sqrt{7} )} \\ \\ = > \frac{8 - 3 \sqrt{7} }{(8) {}^{2} - (3 \sqrt{7} ) {}^{2} } \\ \\ = > \frac{8 - 3 \sqrt{7} }{64 - 63} \\ \\ = > 8 - 3 \sqrt{7} \\ \\ = > \frac{1}{x} = 8 - 3 \sqrt{7}$


So, now,
${x}^{2} + \frac{1}{ {x}^{2} } = {x}^{2} + ( \frac{1}{x} ) {}^{2}$


=
$(8 + 3 \sqrt{7} ) {}^{2} + (8 - 3 \sqrt{7} ) {}^{2} \\ \\ = (64 + 63 + 48 \sqrt{7} ) + (64 + 63 - 48 \sqrt{7} ) \\ \\ = 64 + 63 + 64 + 63 \\ \\ = 254$


Answer :- 254