Question

[1+((a-x)/(a+x))^2 ]÷[1-((a-x)/(a+x))^2 ]

Answer 1

$\frac{\left[1+\left(\frac{\left(a-x\right)}{\left(a+x\right)}\right)^2\right]}{\left[1-\left(\frac{\left(a-x\right)}{\left(a+x\right)}\right)^2\right]}$
$\mathrm{Aplicar\:las\:leyes\:de\:los\:exponentes}:\quad \left(\frac{a}{b}\right)^c=\frac{a^c}{b^c}$
$=\frac{1+\frac{\left(a-x\right)^2}{\left(a+x\right)^2}}{1-\frac{\left(a-x\right)^2}{\left(a+x\right)^2}}$
$\mathrm{Simplificar}\:1+\frac{\left(a-x\right)^2}{\left(a+x\right)^2}\:\mathrm{en\:una\:fraccion}:$
$\mathrm{Encontrar\:el\:minimo\:comun\:denominador\:para}\:\frac{1}{1}+\frac{\left(a-x\right)^2}{\left(a+x\right)^2}:\quad \left(a+x\right)^2$
$=\frac{\left(a+x\right)^2+\left(a-x\right)^2}{\left(a+x\right)^2}$
$= \frac{a^2+2ax+x^{2} +a^{2}-2ax+x^2 }{\left(a+x\right)^2}$
$=\frac{2a^2+2x^2}{\left(a+x\right)^2}$

$= \frac{\frac{2a^2+2x^2}{\left(a+x\right)^2}}{1-\frac{\left(a-x\right)^2}{\left(a+x\right)^2}}$
$\mathrm{Simplificar}\:1-\frac{\left(a-x\right)^2}{\left(a+x\right)^2}\:\mathrm{en\:una\:fraccion}:$
$\mathrm{Encontrar\:el\:minimo\:comun\:denominador\:para}\:\frac{1}{1}-\frac{\left(a-x\right)^2}{\left(a+x\right)^2}:\quad \left(a+x\right)^2$
$=\frac{\left(a+x\right)^2-\left(a-x\right)^2}{\left(a+x\right)^2}$
$=\frac{4ax}{\left(a+x\right)^2}$

$=\frac{\frac{2a^2+2x^2}{\left(a+x\right)^2}}{\frac{4ax}{\left(a+x\right)^2}}$
$\mathrm{Dividir\:fracciones}:\quad \frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a\cdot \:d}{b\cdot \:c}$
$=\frac{\left(2a^2+2x^2\right)\left(a+x\right)^2}{\left(a+x\right)^2\cdot \:4ax}$
$\mathrm{Eliminar\:los\:terminos\:comunes:}\:\left(a+x\right)^2$
$=\frac{2a^2+2x^2}{4ax}$
$\mathrm{Factorizar\:el\:termino\:comun\:}2:$
$=\frac{2\left(a^2+x^2\right)}{4ax}$
$\mathrm{Eliminar\:los\:terminos\:comunes:}\:2$
$=\frac{a^2+x^2}{2ax}$