If $2^x = \sqrt[3]{32}$, then $x$ is equal to

### Concept & Formula This problem is a fundamental test of translating radical notation (roots) into fractional exponents. Once the root is a fraction, you simply express the internal number as a prime base to equate both sides. The core conversion rule is: $$\sqrt[n]{x^m} = x^{\frac{m}{n}}$$ ### Step-by-Step Solution * **Given:** $$2^x = \sqrt[3]{32}$$ * **Calculation:** First, evaluate the number inside the cube root. We need to express $32$ as a power of the base $2$. We know that $2 \times 2 \times 2 \times 2 \times 2 = 32$. Therefore, $32 = 2^5$. Substitute this back into the radical expression: $$2^x = \sqrt[3]{2^5}$$ Convert the cube root into a fractional exponent. A cube root corresponds to the denominator $3$ in a fractional power: $$2^x = (2^5)^{\frac{1}{3}}$$ Multiply the exponents using the power of a power rule: $$2^x = 2^{\frac{5}{3}}$$ Since the bases on both sides of the equation are now exactly $2$, we can drop the bases and directly equate the exponents: $$x = \frac{5}{3}$$ ### Exam Strategy & Shortcut **Direct Translation:** Memorize the prime powers up to $2^6$ and $3^5$. As soon as you read $\sqrt[3]{32}$, you should mentally translate $32$ to $2^5$, and the cube root to a division by $3$ on the exponent, immediately yielding $\frac{5}{3}$. This is a 5-second mental math problem. ### Common Pitfall A frequent trap is confusing the numerator and denominator when converting roots. Students sometimes mistakenly write $(2^5)^{\frac{1}{3}}$ as $2^{\frac{3}{5}}$, leading them to confidently pick the incorrect option (c). Remember: Power is on top, Root is on the bottom. ### Final Answer **Therefore, the correct answer is 5/3.**