First, we need to know the following properties:
[tex]\begin{gathered} x^{-a}=\frac{1}{x^a}^{} \\ \frac{x^a}{x^b}=x^{a-b} \end{gathered}[/tex]
Then, we can rewrite the expression as:
[tex]\begin{gathered} (\frac{3\cdot a^{-2}\cdot b^6}{2\cdot a^{-1}\cdot b^5})^2=(\frac{3}{2}\cdot a^{-2-(-1)}\cdot b^{6-5})^2 \\ (\frac{3\cdot a^{-2}\cdot b^6}{2\cdot a^{-1}\cdot b^5})^2=(\frac{3}{2}a^{-1}\cdot b^1)^2 \\ (\frac{3\cdot a^{-2}\cdot b^6}{2\cdot a^{-1}\cdot b^5})^2=(\frac{3b}{2a})^2 \\ (\frac{3\cdot a^{-2}\cdot b^6}{2\cdot a^{-1}\cdot b^5})^2=\frac{9b^2}{4a^2} \end{gathered}[/tex]
Now, we can replace a by 3 and b by -2 and get:
[tex]\frac{9b^2}{4a^2}=\frac{9\cdot(-2)^2}{4\cdot(3)^2}=\frac{9\cdot4}{4\cdot9}=1[/tex]
Answer: 1
