Let's solve the given problem:

We are asked to compute the value of $27^{-\frac{2}{3}}$ and express the answer as a whole number or a simplified fraction.

Step-by-Step Solution:

1. Rewrite the expression: The exponent $-\frac{2}{3}$ can be interpreted in two parts:

   • The negative sign indicates taking the reciprocal.
   • The fraction $\frac{2}{3}$ implies a root and a power: $27^{\frac{2}{3}} = \left(27^{\frac{1}{3}}\right)^2$.

2. Find the cube root of 27: The cube root of 27 is: $27^{\frac{1}{3}} = 3$

3. Square the result: Raise $3$ to the power of 2: $\left(27^{\frac{1}{3}}\right)^2 = 3^2 = 9$

4. Apply the negative exponent: Take the reciprocal because of the negative exponent: $27^{-\frac{2}{3}} = \frac{1}{9}$

Final Answer:

$\boxed{\frac{1}{9}}$

Do you have any questions or want further clarification?

Follow-Up Questions:

1. What is the general rule for solving fractional exponents?
2. How would $27^{\frac{2}{3}}$ differ from $27^{-\frac{2}{3}}$?
3. Can this method be applied to non-integer bases, e.g., $64^{-\frac{3}{2}}$?
4. Why do we handle the negative exponent last?
5. How does this concept connect to logarithms?

Tip:

To simplify fractional exponents, always deal with roots first (denominator of the fraction) before applying powers (numerator).