Answer
$x=-\frac{5}{3}$
Work Step by Step
$(\sqrt[5] 7)^{x}=7^{2x+3}$
This can be written as
$(7^{1/5})^{x}=7^{2x+3}$
Using the property $(a^{m})^{n}=a^{mn}$, we have
$7^{\frac{x}{5}}=7^{2x+3}$
Equating the exponents, we get
$\frac{x}{5}=2x+3$
Subtracting $\frac{x}{5}$ from both the sides, we obtain
$0=2x-\frac{x}{5}+3$
Or $\frac{9}{5}x+3=0$
Subtracting $3$ from both sides, we get
$\frac{9}{5}x=-3$
Now, multiply by $\frac{5}{9}$ on both sides to get the solution as
$x=-3\times\frac{5}{9}=-\frac{15}{9}=-\frac{5}{3}$
