The value of $(-\frac{1}{216})^{-\frac{2}{3}}$ is

### Concept & Formula This problem tests the rules of negative and fractional indices. A negative exponent indicates the reciprocal of the base, and a fractional exponent indicates a root and a power. $$ (\frac{a}{b})^{-n} = (\frac{b}{a})^n $$ $$ x^{\frac{m}{n}} = (\sqrt[n]{x})^m $$ ### Step-by-Step Solution * **Given:** The expression to evaluate is $(-\frac{1}{216})^{-\frac{2}{3}}$. * **Calculation:** 1. Apply the negative exponent rule by taking the reciprocal of the base fraction to make the exponent positive: $(-216)^{\frac{2}{3}}$ 2. Identify the base as a perfect cube. We know that $(-6) \times (-6) \times (-6) = -216$. $(-6)^3 = -216$ 3. Substitute this back into the expression: $((-6)^3)^{\frac{2}{3}}$ 4. Apply the power of a power rule $(x^a)^b = x^{ab}$ by multiplying the exponents: $3 \times \frac{2}{3} = 2$ 5. The expression simplifies to: $(-6)^2$ 6. Calculate the final square: $36$ ### Exam Strategy & Shortcut When you see a denominator of $3$ in the power, immediately look for a cube root. The cube root of $-216$ is $-6$. Then look at the numerator of the power, which is $2$. Squaring $-6$ gives $36$. Because the power is negative, you flip the original fraction $\frac{1}{216}$ to $216$ first. The sequence of thought is: Flip to $216$, cube root to $6$, square to $36$. ### Common Pitfall Students often drop the negative sign inside the parenthesis too early or assume the negative exponent makes the final answer negative. Remember that squaring any real number, including negative bases like $-6$, results in a positive value. ### Final Answer **Therefore, the correct answer is 36.**