The value of $27^{-\frac{2}{3}}$ lies between

### Concept & Formula This problem tests the application of negative and fractional exponents. A negative exponent denotes the reciprocal of the base, and a fractional exponent represents a root and a power. $$ x^{-n} = \frac{1}{x^n} $$ $$ x^{\frac{m}{n}} = (\sqrt[n]{x})^m $$ ### Step-by-Step Solution * **Given:** The expression to evaluate is $27^{-\frac{2}{3}}$. * **Calculation:** 1. First, express the base $27$ as a prime power. We know that $27 = 3^3$. 2. Substitute this back into the expression: $(3^3)^{-\frac{2}{3}}$ 3. Apply the power of a power rule $(x^a)^b = x^{ab}$ by multiplying the exponents: $3 \times (-\frac{2}{3}) = -2$ 4. The expression simplifies to: $3^{-2}$ 5. Apply the negative exponent rule to take the reciprocal: $\frac{1}{3^2}$ 6. Calculate the final fraction: $\frac{1}{9}$ 7. Determine the range. The value $\frac{1}{9}$ (approximately $0.111$) is a proper fraction that is greater than $0$ but less than $1$. ### Exam Strategy & Shortcut Look at the denominator of the exponent ($3$). This means you need the cube root of the base ($27$), which is $3$. Look at the numerator with its sign ($-2$). You must raise $3$ to the power of $-2$, giving $1/9$. You don't need to calculate the exact decimal to know that $1/9$ is between $0$ and $1$. The entire mental process should take less than 5 seconds. ### Common Pitfall A common error is to mistakenly bring the negative sign to the front (e.g., $-27^{\frac{2}{3}} = -9$) or to evaluate the fraction upside down. Always remember that a negative exponent affects the position of the number (numerator vs. denominator), not its positive/negative parity. ### Final Answer **Therefore, the correct answer is 0 and 1.**