Solve: \(\left(\sqrt{\frac{3}{5}}\right)^{x+1} = \frac{125}{27}\)
Solution
\[ \left(\sqrt{\frac{3}{5}}\right)^{x+1} = \frac{125}{27} \]
\[ \Rightarrow \left(\frac{3}{5}\right)^{\frac{x+1}{2}} = \frac{5^3}{3^3} \]
\[ \Rightarrow \frac{3^{\frac{x+1}{2}}}{5^{\frac{x+1}{2}}} = \frac{5^3}{3^3} \]
\[ \Rightarrow 3^{\frac{x+1}{2}} \cdot 3^3 = 5^{\frac{x+1}{2}} \cdot 5^3 \]
\[ \Rightarrow 3^{\frac{x+1}{2} + 3} = 5^{\frac{x+1}{2} + 3} \]
\[ \Rightarrow \frac{x+1}{2} + 3 = 0 \]
\[ \Rightarrow \frac{x+1}{2} = -3 \]
\[ \Rightarrow x+1 = -6 \]
\[ \Rightarrow x = -7 \]
Final Answer:
\[ \boxed{x = -7} \]