To determine the value of the expression \(\left(3 x^3 y^{-2}\right)^2\) for \(x = -2\) and \(y = 5\), let's go through the problem step-by-step.
### Step 1: Substitute the Values
First, substitute \(x = -2\) and \(y = 5\) into the expression:
[tex]\[
\left(3 (-2)^3 5^{-2}\right)^2
\][/tex]
### Step 2: Evaluate Inside the Parentheses
Evaluate the individual terms inside the parentheses:
- Calculate \((-2)^3\):
[tex]\[
(-2)^3 = -8
\][/tex]
- Calculate \(5^{-2}\) which is the same as \(\frac{1}{5^2}\):
[tex]\[
5^{-2} = \frac{1}{25}
\][/tex]
### Step 3: Combine the Results
Next, multiply these results together along with the constant coefficient 3:
[tex]\[
3 \cdot (-8) \cdot \frac{1}{25} = 3 \cdot -8 \cdot 0.04 = -0.96
\][/tex]
So inside the parentheses we have:
[tex]\[
(3 (-2)^3 5^{-2}) = -0.96
\][/tex]
### Step 4: Square the Result
We now square the result obtained in the previous step:
[tex]\[
(-0.96)^2 = 0.9216
\][/tex]
### Step 5: Identify the Correct Choice
We need to find which of the given multiple-choice options match the numeric result of \(0.9216\). Let's simplify each option step-by-step to see which equals the computed result:
1. \(\frac{3^2(-2)^6}{5^4}\):
[tex]\[
3^2 = 9
\][/tex]
[tex]\[
(-2)^6 = 64
\][/tex]
[tex]\[
5^4 = 625
\][/tex]
[tex]\[
\frac{9 \cdot 64}{625} = \frac{576}{625} = 0.9216
\][/tex]
2. \(\frac{3(-2)^6}{5^4}\):
[tex]\[
3 \cdot 64 = 192
\][/tex]
[tex]\[
\frac{192}{625} \approx 0.3072 \neq 0.9216
\][/tex]
3. \(\frac{3^2(5)^6}{(-2)^4}\):
[tex]\[
3^2 = 9
\][/tex]
[tex]\[
5^6 = 15625
\][/tex]
[tex]\[
(-2)^4 = 16
\][/tex]
[tex]\[
\frac{9 \cdot 15625}{16} = 87812.5 \neq 0.9216
\][/tex]
4. \(\frac{3}{(2)^6 5^4}\):
[tex]\[
(2)^6 = 64
\][/tex]
[tex]\[
5^4 = 625
\][/tex]
[tex]\[
64 \cdot 625 = 40000
\][/tex]
[tex]\[
\frac{3}{40000} = 0.000075 \neq 0.9216
\][/tex]
Thus, the correct choice is:
[tex]\[
\frac{3^2(-2)^6}{5^4}
\][/tex]
So, the final answer is:
[tex]\[
\boxed{\frac{3^2(-2)^6}{5^4}}
\][/tex]
