If $\sqrt{2^n} = 64$, then the value of $n$ is

### Concept & Formula This question requires converting a square root expression into a fractional exponent and mapping a whole number to its prime base power to equate the exponents. The critical translations are: $$\sqrt{x} = x^{\frac{1}{2}}$$ $$(\sqrt{x})^2 = x$$ ### Step-by-Step Solution * **Given:** $$\sqrt{2^n} = 64$$ * **Calculation (Method 1: Fractional Exponents):** Express the square root as a power of $\frac{1}{2}$: $$(2^n)^{\frac{1}{2}} = 64$$ $$2^{\frac{n}{2}} = 64$$ Convert $64$ into a base of $2$: $$64 = 2^6$$ Substitute and equate the bases: $$2^{\frac{n}{2}} = 2^6$$ $$\frac{n}{2} = 6$$ $$n = 12$$ ### Exam Strategy & Shortcut **Squaring Both Sides:** An extremely fast way to bypass fractional exponents is to immediately square both sides of the original equation to eliminate the radical. $$(\sqrt{2^n})^2 = 64^2$$ $$2^n = (2^6)^2$$ $$2^n = 2^{12}$$ $$n = 12$$ ### Common Pitfall A very frequent mistake is identifying $64$ as $2^6$ and then immediately assuming $n = 6$, entirely ignoring the square root over the $2^n$ term. Always isolate the variable entirely. ### Final Answer **Therefore, the correct answer is 12.**