Question

all expressions equivalent to the given expression
(3^3 x 3^(-2))^-2

Answer 1

To simplify the expression \((3^3 \cdot 3^{-2})^{-2}\), we can follow these steps:

1. **Combine the exponents inside the parentheses**:
\[
3^3 \cdot 3^{-2} = 3^{3 + (-2)} = 3^{3 - 2} = 3^1 = 3
\]

2. **Now raise that result to the power of -2**:
\[
(3^1)^{-2} = 3^{1 \cdot (-2)} = 3^{-2}
\]

3. **Finally, we can express this with a positive exponent**:
\[
3^{-2} = \frac{1}{3^2} = \frac{1}{9}
\]

So, the expression \((3^3 \cdot 3^{-2})^{-2}\) simplifies to:

\[
3^{-2} \quad \text{or} \quad \frac{1}{9}
\]

### Equivalent Expressions:
Here are some equivalent forms of the original expression:
1. \(3^{-2}\)
2. \(\frac{1}{9}\)
3. \((3^3 \cdot 3^{-2})^{-2}\) (the original expression)
4. \(\frac{(3^3)}{(3^{2})^2}\) (using the property of exponents)
5. \(\frac{27}{9^2}\) (substituting \(3^3 = 27\) and \(3^{-2} = \frac{1}{9}\))

These are various valid forms representing the same value.