Question

Find the value of $x$ if ${\left(\frac{5}{7}\right)}^6{\left(\frac{7}{5}\right)}^{-9}={\left(\frac{5}{7}\right)}^{3x}$

Answer 1

Compute the required value:

$$\begin{gathered} \Rightarrow {\left(\frac{5}{7}\right)}^6{\left(\frac{7}{5}\right)}^{-9}={\left(\frac{5}{7}\right)}^{3x} \\ \Rightarrow {\left(\frac{5}{7}\right)}^6{\left(\frac{5}{7}\right)}^9={\left(\frac{5}{7}\right)}^{3x}[\because {\left[\frac{a}{b}\right]}^{-m}={\left[\frac{b}{a}\right]}^m] \\ \Rightarrow {\left(\frac{5}{7}\right)}^{6+9}={\left(\frac{5}{7}\right)}^{3x} \\ \Rightarrow 15=3x \\ \Rightarrow x=5 \end{gathered}$$

Hence, the value of $x$ is $5$.