Certainly! Let's go through the solution of the given expression step by step:
We have the expression:
[tex]\[
\left(3 x^3 y^{-2}\right)^2
\][/tex]
where [tex]\( x = -2 \)[/tex] and [tex]\( y = 5 \)[/tex].
### Step-by-Step Solution:
1. Calculate [tex]\( x^3 \)[/tex] and [tex]\( y^{-2} \)[/tex]:
- [tex]\( x = -2 \)[/tex], so [tex]\( x^3 = (-2)^3 = -8 \)[/tex].
- [tex]\( y = 5 \)[/tex], so [tex]\( y^{-2} = \frac{1}{y^2} = \frac{1}{5^2} = \frac{1}{25} = 0.04 \)[/tex].
2. Combine the expression [tex]\( 3 x^3 y^{-2} \)[/tex]:
- First, calculate [tex]\( 3 x^3 \)[/tex]:
[tex]\[
3 x^3 = 3 \cdot (-8) = -24
\][/tex]
- Now, multiply by [tex]\( y^{-2} \)[/tex]:
[tex]\[
3 x^3 y^{-2} = -24 \cdot 0.04 = -0.96
\][/tex]
3. Square the combined result:
- We now have:
[tex]\[
\left(-0.96\right)^2 = 0.9216
\][/tex]
So the simplified value of the expression when [tex]\( x = -2 \)[/tex] and [tex]\( y = 5 \)[/tex] is [tex]\( 0.9216 \)[/tex].
### Evaluating Given Options:
Let's check the given options with our calculated values:
1. [tex]\(\frac{3^2(-2)^6}{5^4}\)[/tex]
- Calculate:
[tex]\[
(3^2 = 9), \; (-2)^6 = 64, \; (5^4 = 625)
\][/tex]
- Hence:
[tex]\[
\frac{9 \cdot 64}{625} = \frac{576}{625} = 0.9216
\][/tex]
- This matches our calculated value of [tex]\( 0.9216 \)[/tex].
2. [tex]\(\frac{3(-2)^6}{5^4}\)[/tex]
- Calculate:
[tex]\[
3 \cdot 64 = 192
\][/tex]
- Hence:
[tex]\[
\frac{192}{625}
\][/tex]
- This does not match our calculated value.
3. [tex]\(\frac{3^2(5)^6}{(-2)^4}\)[/tex]
- This option's calculation is quite different from what we need, and evaluating would give a completely different number.
4. [tex]\(\frac{3}{(2)^6 5^4}\)[/tex]
- Calculate:
[tex]\[
2^6 = 64
\][/tex]
[tex]\[
3 / (64 \cdot 625)
\][/tex]
- This fraction would be much smaller than our calculated value.
Conclusion:
The correct option that matches our calculated value is:
[tex]\(\frac{3^2(-2)^6}{5^4}\)[/tex]
So, the correct option is the first one.
