Correct. Because both sides of the equation have expressions raised to the same exponent, the bases must be equal.

$\displaystyle{{\left({x}+{4}\right)}}^{{7}}={{9}}^{{7}}\usepackage{cmbright}\Huge\$

$\displaystyle{x}+{4}={9}\usepackage{cmbright}\Huge\$

$\displaystyle{x}={5}\usepackage{cmbright}\Huge\$