Question

Represent the following mixed infinite decimal periodic fractions as common fractions:
$(x+\sqrt{x^2-1}{)}^2+(x+\sqrt{x^2-1}{)}^{-2}+2(1-2x^2)$

Answer 1

$(x+\sqrt{x^2-1})+(x+\sqrt{x^2-1}{)}^{-2}+2(1-2x^2)$

$=(x+{\sqrt{x^2-1)}}^{-1}$ can be simplified

as $\frac{1}{x+\sqrt{x^2-1}}$

$=\frac{x-\sqrt{x^2-1}}{(x{)}^2-(\sqrt{x^2-1}{)}^2}$

$=\frac{x-\sqrt{x^2-1}}{x^2-x^2+1}=x-\sqrt{x^2-1}$

So,

$(x+\sqrt{x^2-1}{)}^2+(x-\sqrt{x^2-1}{)}^2+2-4x^2$

$=(x+\sqrt{x^2-1}{)}^2+(x-\sqrt{x^2-1}{)}^2+2(x+\sqrt{x^2-1})(x-\sqrt{x^2-1})-4x^2$

$=(x+\sqrt{x^2-1}+x-\sqrt{x^2-1}{)}^2-4x^2$

$=(2x{)}^2-4x^2$

$=4x^2-4x^2=0$