If $\sqrt{3^n} = 729$, then the value of $n$ is

Concept & Formula To solve equations with the unknown variable in the exponent, express both sides of the equation using the exact same base so the exponents can be directly equated. $$(a^x)^y = a^{xy}$$ Step-by-Step Solution * **Given:** $\sqrt{3^n} = 729$ * **Calculation:** Express the square root as a fractional exponent on the left side: $(3^n)^{\frac{1}{2}} = 3^{\frac{n}{2}}$ * Express the right side ($729$) as a power of the base $3$: $729 = 9 \times 81 = 3^2 \times 3^4 = 3^6$ * Rewrite the full equation with the matching bases: $3^{\frac{n}{2}} = 3^6$ * Since the bases are identical ($3$), their exponents must be equal: $\frac{n}{2} = 6$ * Multiply by $2$ to solve for $n$: $n = 12$ Exam Strategy & Shortcut If you know your powers of $3$, you know $3^6 = 729$. Since the left side is a square root, the internal power $n$ must be double the required exponent to survive the root process. Therefore, $n = 6 \times 2 = 12$. Common Pitfall Equating $n$ directly to $6$ after finding that $3^6 = 729$, completely ignoring the effect of the square root radical on the left side. Final Answer **Therefore, the correct answer is 12.**