To determine which expression is equivalent to [tex]\(\left(4 x^3 y^5\right)\left(3 x^5 y\right)^2\)[/tex], let's break it down step by step.
1. Simplify the expression inside the parentheses:
[tex]\(\left(3 x^5 y\right)^2\)[/tex]
When an expression is squared, you raise each part of the expression to the power of two. Thus:
[tex]\[
(3 x^5 y)^2 = 3^2 \cdot \left(x^5\right)^2 \cdot y^2
\][/tex]
Calculating each part:
- [tex]\(3^2 = 9\)[/tex]
- [tex]\(\left(x^5\right)^2 = x^{10}\)[/tex] (because [tex]\((a^m)^n = a^{mn}\)[/tex])
- [tex]\(y^2\)[/tex]
So:
[tex]\[
\left(3 x^5 y\right)^2 = 9 x^{10} y^2
\][/tex]
2. Multiply this result by the other term:
Now we need to multiply this by [tex]\(4 x^3 y^5\)[/tex]:
[tex]\[
\left(4 x^3 y^5\right) \cdot \left(9 x^{10} y^2\right)
\][/tex]
We can combine the coefficients and the powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] separately:
- Coefficients: [tex]\(4 \cdot 9 = 36\)[/tex]
- Powers of [tex]\(x\)[/tex]: [tex]\(x^3 \cdot x^{10} = x^{3+10} = x^{13}\)[/tex]
- Powers of [tex]\(y\)[/tex]: [tex]\(y^5 \cdot y^2 = y^{5+2} = y^7\)[/tex]
So:
[tex]\[
\left(4 x^3 y^5\right) \cdot \left(9 x^{10} y^2\right) = 36 x^{13} y^7
\][/tex]
3. Check the answer choices:
- [tex]\(24 x^{13} y^7\)[/tex]
- [tex]\(36 x^{13} y^7\)[/tex]
- [tex]\(36 x^{20} y^6\)[/tex]
- [tex]\(144 x^{16} y^{12}\)[/tex]
The correct expression is [tex]\(36 x^{13} y^7\)[/tex].
Therefore, the expression that is equivalent to [tex]\(\left(4 x^3 y^5\right)\left(3 x^5 y\right)^2\)[/tex] is [tex]\(\boxed{36 x^{13} y^7}\)[/tex].
