Question

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

Answer 1

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

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

$=\frac{(\sqrt{x}{)}^2-1}{(\sqrt{x}{)}^3-1}\times \sqrt{x}(x^{\frac{3}{2}}-1)$

$=\frac{(x-1)}{x^{\frac{3}{2}}-1}\sqrt{x}(x^{\frac{3}{2}}-1)$

$=\sqrt{x}(x-1)$