If $289 = 17^{\frac{1}{5}x}$, then $x$ is

## Concept & Formula This problem requires solving an exponential equation by equating the bases. If two exponential terms with the same base are equal, their exponents must also be equal. $$a^m = a^n \implies m = n$$ ## Step-by-Step Solution * **Given:** The equation is $289 = 17^{\frac{1}{5}x}$. * **Calculation / Deduction:** * First, express the left-hand side ($289$) in terms of the base on the right-hand side ($17$). * We know that $17 \times 17 = 289$, so $289 = 17^2$. * Substitute this back into the equation: $$17^2 = 17^{\frac{x}{5}}$$ * Since the bases ($17$) are now identical on both sides, we can equate the exponents: $$2 = \frac{x}{5}$$ * Multiply both sides by $5$ to isolate $x$: $$x = 2 \times 5$$ $$x = 10$$ * Looking at the given options: (a) $\frac{2}{5}$, (b) $8$, (c) $16$, (d) $32$. Since $10$ is not listed, the answer is "None of these". ## Exam Strategy & Shortcut Memorize squares up to $30$. Recognizing immediately that $289$ is $17^2$ allows you to skip writing the base and directly solve the exponent equation $2 = \frac{x}{5} \implies x = 10$ in your head in under 10 seconds. ## Common Pitfall A common error is mishandling the fractional exponent during algebraic manipulation. A student might incorrectly divide by $5$ instead of multiplying, arriving at $x = \frac{2}{5}$, which is a trap answer provided in option (a). ## Final Answer **Therefore, the correct answer is None of these.**