(Laws of Exponents with Integer Exponents LC) Which is an equivalent expression for \( \left(\left(\frac{3}{7}\right)^{2} \cdot\left(\frac{2}{3}\right)^{-3}\right)^{-1} \) ? \( \left(\frac{7}{3}\right)^{-2} \cdot\left(\frac{2}{3}\right)^{-3} \) \( \left(\frac{7}{3}\right)^{2} \cdot\left(\frac{2}{3}\right)^{3} \) \( \left(\frac{3}{7}\right) \cdot\left(\frac{2}{3}\right)^{-4} \) \( \left(\frac{3}{7}\right)^{2} \cdot\left(\frac{2}{3}\right)^{-3} \)

To determine which expression is equivalent to \[ \left(\left(\frac{3}{7}\right)^{2} \cdot \left(\frac{2}{3}\right)^{-3}\right)^{-1}, \] let's simplify step by step using the **Laws of Exponents**. 1. **Simplify the Inner Expression:** Start with the inner part: \[ \left(\frac{3}{7}\right)^{2} \cdot \left(\frac{2}{3}\right)^{-3} \] Recall that \( \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n} \). Applying this: \[ \left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^{3} \] So the expression becomes: \[ \left(\frac{3}{7}\right)^{2} \cdot \left(\frac{3}{2}\right)^{3} \] 2. **Combine the Exponents:** \[ \left(\frac{3}{7}\right)^{2} \cdot \left(\frac{3}{2}\right)^{3} = \frac{3^{2} \cdot 3^{3}}{7^{2} \cdot 2^{3}} = \frac{3^{5}}{7^{2} \cdot 2^{3}} \] 3. **Apply the Outer Exponent (-1):** \[ \left(\frac{3^{5}}{7^{2} \cdot 2^{3}}\right)^{-1} = \frac{7^{2} \cdot 2^{3}}{3^{5}} \] 4. **Compare with the Given Options:** Let's analyze each option: - **Option 1:** \[ \left(\frac{7}{3}\right)^{-2} \cdot \left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{7}\right)^{2} \cdot \left(\frac{3}{2}\right)^{3} = \frac{3^{5}}{7^{2} \cdot 2^{3}} \] This is the reciprocal of our simplified expression, not equivalent to the original expression. - **Option 2:** \[ \left(\frac{7}{3}\right)^{2} \cdot \left(\frac{2}{3}\right)^{3} = \frac{7^{2} \cdot 2^{3}}{3^{5}} = \frac{392}{243} \] This exactly matches our simplified expression. - **Option 3:** \[ \left(\frac{3}{7}\right) \cdot \left(\frac{2}{3}\right)^{-4} = \frac{3}{7} \cdot \left(\frac{3}{2}\right)^{4} = \frac{243}{112} \] This does not match our simplified expression. - **Option 4:** \[ \left(\frac{3}{7}\right)^{2} \cdot \left(\frac{2}{3}\right)^{-3} = \frac{3^{5}}{7^{2} \cdot 2^{3}} \] Similar to Option 1, this is the reciprocal of our target expression. **Therefore, the correct equivalent expression is:** \[ \boxed{\left(\frac{7}{3}\right)^{2} \cdot\left(\frac{2}{3}\right)^{3}} \]