Certainly! Let's break down the provided expressions to see which one correctly computes the value of the given expression [tex]\(\left(3 x^3 y^{-2}\right)^2\)[/tex] when [tex]\(x = -2\)[/tex] and [tex]\(y = 5\)[/tex].
First, let's start with the expression:
[tex]\[
\left(3 x^3 y^{-2}\right)^2
\][/tex]
We'll substitute [tex]\(x = -2\)[/tex] and [tex]\(y = 5\)[/tex]:
[tex]\[
\left(3 (-2)^3 5^{-2}\right)^2
\][/tex]
Calculate each part inside the parentheses:
1. [tex]\( (-2)^3 = -8 \)[/tex]
2. [tex]\( 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \)[/tex]
Now, substituting these values into the expression:
[tex]\[
\left(3 \cdot (-8) \cdot \frac{1}{25}\right)^2
\][/tex]
[tex]\[
\left(\frac{3 \cdot (-8)}{25}\right)^2
\][/tex]
[tex]\[
\left(\frac{-24}{25}\right)^2
\][/tex]
Now, square the fraction:
[tex]\[
\left(\frac{-24}{25}\right)^2 = \frac{(-24)^2}{25^2}
\][/tex]
[tex]\[
= \frac{576}{625}
\][/tex]
This is the simplified form of the original expression. The closest matching option to this simplified form should be verified to see if it equals our simplified result of [tex]\(\frac{576}{625}\)[/tex].
Let's evaluate the given options:
1. [tex]\(\frac{3^2 \cdot (-2)^6}{5^4}\)[/tex]:
[tex]\[
= \frac{9 \cdot 64}{625} = \frac{576}{625}
\][/tex]
2. [tex]\(\frac{3 \cdot (-2)^6}{5^4}\)[/tex]:
[tex]\[
= \frac{3 \cdot 64}{625} = \frac{192}{625}
\][/tex]
3. [tex]\(\frac{3^2 \cdot 5^6}{(-2)^4}\)[/tex]:
[tex]\[
= \frac{9 \cdot 15625}{16} = 8789.0625
\][/tex]
4. [tex]\(\frac{3}{(2^6 \cdot 5^4)}\)[/tex]:
[tex]\[
= \frac{3}{4096 \cdot 625} = \frac{3}{2560000} = 7.5 \times 10^{-5}
\][/tex]
From these calculations, we can see that the correct expression that matches our original calculation, [tex]\(\frac{576}{625}\)[/tex], is:
[tex]\[
\boxed{\frac{3^2(-2)^6}{5^4}}
\][/tex]
