Sure, let's solve the given expression step-by-step.
We are given the expression:
[tex]\[
\frac{-2 a^{-3} b^2}{a}
\][/tex]
and we need to evaluate it for [tex]\( a = -2 \)[/tex] and [tex]\( b = 3 \)[/tex].
### Step 1: Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]
First, substitute [tex]\( a = -2 \)[/tex] and [tex]\( b = 3 \)[/tex] into the expression.
[tex]\[
\frac{-2 (-2)^{-3} (3)^2}{-2}
\][/tex]
### Step 2: Simplify the exponents
Calculate each part of the expression involving an exponent.
For [tex]\( (-2)^{-3} \)[/tex]:
[tex]\[
(-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} = -\frac{1}{8}
\][/tex]
For [tex]\( (3)^2 \)[/tex]:
[tex]\[
(3)^2 = 9
\][/tex]
Now, the expression is simplified to:
[tex]\[
\frac{-2 \left(-\frac{1}{8}\right) (9)}{-2}
\][/tex]
### Step 3: Multiply the constants and fractions
First, multiply [tex]\(-2\)[/tex], [tex]\(-\frac{1}{8}\)[/tex], and [tex]\(9\)[/tex]:
[tex]\[
-2 \times -\frac{1}{8} = \frac{2}{8} = \frac{1}{4}
\][/tex]
Then:
[tex]\[
\frac{1}{4} \times 9 = \frac{9}{4}
\][/tex]
So we now have:
[tex]\[
\frac{\frac{9}{4}}{-2}
\][/tex]
### Step 4: Divide by [tex]\(-2\)[/tex]
To divide [tex]\(\frac{9}{4}\)[/tex] by [tex]\(-2\)[/tex], multiply by the reciprocal of [tex]\(-2\)[/tex] (which is [tex]\(-\frac{1}{2}\)[/tex]):
[tex]\[
\frac{9}{4} \times -\frac{1}{2} = -\frac{9}{8}
\][/tex]
### Conclusion
Thus, the result of evaluating the expression [tex]\(\frac{-2 a^{-3} b^2}{a}\)[/tex] for [tex]\(a = -2\)[/tex] and [tex]\(b = 3\)[/tex] is:
[tex]\[
\boxed{-\frac{9}{8}}
\][/tex]
So, the correct answer is option D) [tex]\(-\frac{9}{8}\)[/tex].
