2003 AMC 10B Problem 9

Convert every factor to base $5$. Since $25 = 5^2$,
$$25^{-2} = 5^{-4}, \qquad 25^{17/x} = 5^{34/x}.$$

The right side is then a single power of $5$ whose exponent is the top exponent minus the two bottom exponents:
$$\frac{5^{48/x}}{5^{26/x} \cdot 5^{34/x}} = 5^{(48 - 26 - 34)/x} = 5^{-12/x}.$$

Equal powers of the same base have equal exponents, so
$$-4 = -\frac{12}{x} \quad\Longrightarrow\quad x = 3.$$

The answer is $\boxed{\textbf{(B)}\ 3}$.

An exponential equation with a single underlying base is really a linear equation in the exponents. The only skills involved are recognizing $25 = 5^2$ and applying $a^m / (a^n a^p) = a^{m-n-p}$ cleanly. Convert to a common base first; never try to compute the numbers themselves.

• Forgetting that $25^{17/x}$ is $5^{34/x}$, not $5^{17/x}$, which changes the exponent to $5/x$ and gives no listed answer.
• Subtracting only one of the denominator exponents, or writing $25^{-2}$ as $5^{-2}$.