The value of $(\sqrt{8})^{\frac{1}{3}}$ is

### Concept & Formula This question requires converting a radical (square root) into a fractional exponent, and then applying the power of a power rule. $$ \sqrt[n]{x} = x^{\frac{1}{n}} $$ $$ (x^a)^b = x^{ab} $$ ### Step-by-Step Solution * **Given:** The expression to evaluate is $(\sqrt{8})^{\frac{1}{3}}$. * **Calculation:** 1. Convert the square root of $8$ into a fractional exponent: $\sqrt{8} = 8^{\frac{1}{2}}$ 2. Substitute this back into the original expression: $(8^{\frac{1}{2}})^{\frac{1}{3}}$ 3. Apply the power of a power rule by multiplying the fractions: $8^{(\frac{1}{2} \times \frac{1}{3})} = 8^{\frac{1}{6}}$ 4. Convert the base $8$ into a prime base to simplify further. We know $8 = 2^3$. $(2^3)^{\frac{1}{6}}$ 5. Multiply the exponents again: $2^{(3 \times \frac{1}{6})} = 2^{\frac{3}{6}}$ 6. Simplify the fraction in the exponent: $2^{\frac{1}{2}}$ 7. Convert the fractional exponent back to a radical for the final answer: $\sqrt{2}$ ### Exam Strategy & Shortcut Reverse the order of operations for speed. The power $\frac{1}{3}$ is a cube root. The expression is effectively the cube root of the square root of $8$. Because $8$ is a perfect cube ($2^3$), handle the cube root first! The cube root of $8$ is $2$. You are left with just the square root over that $2$, resulting instantly in $\sqrt{2}$. ### Common Pitfall Students often struggle with nested radicals and fractional powers. A common mistake is to multiply the base by the exponent (e.g., $8 \times \frac{1}{3}$) instead of recognizing it as a root. Always convert roots to fractions if you are unsure how to proceed. ### Final Answer **Therefore, the correct answer is $\sqrt{2}$.**