To solve the given expression when [tex]\( a = -2 \)[/tex] and [tex]\( b = -3 \)[/tex], we need to evaluate it step by step:
The expression is:
[tex]\[
\left(\frac{3 a^{-3} b^2}{2 a^{-1} b^0}\right)^2
\][/tex]
We'll break it down into parts:
1. Evaluate the numerator [tex]\( 3 a^{-3} b^2 \)[/tex]:
- First, compute [tex]\( a^{-3} \)[/tex]:
[tex]\[
a^{-3} = (-2)^{-3} = -\frac{1}{8}
\][/tex]
- Compute [tex]\( b^2 \)[/tex]:
[tex]\[
b^2 = (-3)^2 = 9
\][/tex]
- Multiply these results with 3:
[tex]\[
3 a^{-3} b^2 = 3 \times \left(-\frac{1}{8}\right) \times 9 = 3 \times -\frac{9}{8} = -\frac{27}{8} = -3.375
\][/tex]
2. Evaluate the denominator [tex]\( 2 a^{-1} b^0 \)[/tex]:
- First, compute [tex]\( a^{-1} \)[/tex]:
[tex]\[
a^{-1} = (-2)^{-1} = -\frac{1}{2}
\][/tex]
- Compute [tex]\( b^0 \)[/tex]:
[tex]\[
b^0 = 1
\][/tex]
- Multiply these results with 2:
[tex]\[
2 a^{-1} b^0 = 2 \times \left(-\frac{1}{2}\right) \times 1 = 2 \times -\frac{1}{2} = -1
\][/tex]
3. Form the fraction/numerator [tex]\( \frac{3 a^{-3} b^2}{2 a^{-1} b^0} \)[/tex]:
[tex]\[
\frac{3 a^{-3} b^2}{2 a^{-1} b^0} = \frac{-3.375}{-1} = 3.375
\][/tex]
4. Square the result:
[tex]\[
\left(\frac{3 a^{-3} b^2}{2 a^{-1} b^0}\right)^2 = (3.375)^2 = 11.390625
\][/tex]
Thus, the value of the given expression when [tex]\( a = -2 \)[/tex] and [tex]\( b = -3 \)[/tex] is:
[tex]\[ 11.390625 \][/tex]
To match this result with the provided options, we note:
[tex]\[
11.390625 = \frac{729}{64}
\][/tex]
so the correct answer is:
[tex]\[
\boxed{\frac{729}{64}}
\][/tex]
