Question

Solve for $x$:

${\left(\frac{2}{5}\right)}^{2x-2}=\frac{32}{3125}$

Answer 1

From question,

${\left(\frac{2}{5}\right)}^{2x-2}=\frac{32}{3125}$

$\Rightarrow {\frac{2}{5}}^{2x-2}=\frac{2^5}{5^5}$

$\Rightarrow {\frac{2}{5}}^{2x-2}={\left(\frac{2}{5}\right)}^5$ $[\because \frac{x^a}{y^a}={\left(\frac{x}{y}\right)}^a]$

$\Rightarrow {\left(\frac{2}{5}\right)}^{2x-2}={\left(\frac{2}{5}\right)}^5$

$\Rightarrow 2x-2=5$ [$\because x^a=x^b⟹a=b$]

$\Rightarrow 2x=5+2=7$

$\Rightarrow x=\frac{7}{2}$

Hence, $x=\frac{7}{2}$