imc

Manipulate the powers of $5$ in order to get a clean expression. $$\frac{5^{\frac{48}{x}}}{5^{\frac{26}{x}}\cdot 25^{\frac{17}{x}}}=\frac{5^{\frac{48}{x}}}{5^{\frac{26}{x}}\cdot 5^{\frac{34}{x}}}=5^{\frac{48}{x}-(\frac{26}{x}+\frac{34}{x})}=5^{-\frac{12}{x}}$$ $$25^{-2}=(5^2{)}^{-2}=5^{-4}$$ $$5^{-4}=5^{-\frac{12}{x}}$$ If two numbers are equal, and their bases are equal, then their exponents are equal as well. Set the two exponents equal to each other. $$\begin{array}{rl}-4& =\frac{-12}{x}\\ -4x& =-12\\ x& =\boxed{3}\end{array}$$