[tex]\displaystyle \sf \text{Propriedades de Potencia\c c\~ao}: \\\\ \sqrt[\sf b]{\sf a^{c}} = a^{c/b} \\\\ a^{b} = a^{c} \to b=c[/tex]

[tex]\displaystyle \sf temos: \\\\ \sqrt[\sf 3]{\sf 7^{x-2}} = \sqrt[\sf x]{7^5} \\\\ 7^{(x-2)/3} = 7^{5/x} \\\\ \frac{x-2}{3} = \frac{5}{x} \\\\\\ x(x-2) = 5.3 \\\\ x^2-2x=15 \\\\ x^2-2x+1 = 15+1 \\\\ (x-1)^2=16 \\\\ \sqrt{(x-1)^2}=\pm\sqrt{16} \\\\ x-1 = \pm 4 \\\\ x = 1\pm 4 \\\\ x = 1+ 4 \to x = 5 \\\\\ x=1-4 \to x = -3 \\\\\ \text{solu\c c\~oes :} \\\\ \large\boxed{\sf \ x= 5 \ \ e \ \ x = - 3\ } \checkmark[/tex]