To solve the problem, we need to simplify the given expression [tex]\(\left(4 x^3 y^5\right)\left(3 x^5 y\right)^2\)[/tex].
### Step-by-Step Solution:
1. Simplify the exponentiation term:
[tex]\[
\left(3 x^5 y\right)^2
\][/tex]
For this step, we need to distribute the exponent of 2 to each factor inside the parentheses:
[tex]\[
\left(3 x^5 y\right)^2 = 3^2 \cdot \left(x^5\right)^2 \cdot y^2
\][/tex]
This simplifies as follows:
[tex]\[
3^2 = 9
\][/tex]
[tex]\[
\left(x^5\right)^2 = x^{5 \cdot 2} = x^{10}
\][/tex]
[tex]\[
y^2 = y^2
\][/tex]
Putting it all together:
[tex]\[
\left(3 x^5 y\right)^2 = 9 x^{10} y^2
\][/tex]
2. Multiply the simplified terms:
Now we need to multiply this result by the other factor in the original expression:
[tex]\[
\left(4 x^3 y^5\right) \left(9 x^{10} y^2\right)
\][/tex]
To do this, we multiply the coefficients and add the exponents of like bases:
[tex]\[
\text{Coefficients: } 4 \cdot 9 = 36
\][/tex]
[tex]\[
\text{Exponents of } x: 3 + 10 = 13
\][/tex]
[tex]\[
\text{Exponents of } y: 5 + 2 = 7
\][/tex]
3. Combine the results:
[tex]\[
\left(4 x^3 y^5\right)\left(9 x^{10} y^2\right) = 36 x^{13} y^7
\][/tex]
Therefore, the expression 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]
