To simplify the expression [tex]\(\left( 4 x^3 y^3 \right) \left( 2 x^2 y \right)\)[/tex], let's break it down step-by-step:
1. Multiply the constants:
The constants in the expression are 4 and 2. Multiplying these gives:
[tex]\[
4 \cdot 2 = 8
\][/tex]
2. Multiply the [tex]\(x\)[/tex] terms:
The [tex]\(x\)[/tex] terms are [tex]\(x^3\)[/tex] and [tex]\(x^2\)[/tex]. When multiplying exponents with the same base, you add the exponents:
[tex]\[
x^3 \cdot x^2 = x^{3+2} = x^5
\][/tex]
3. Multiply the [tex]\(y\)[/tex] terms:
The [tex]\(y\)[/tex] terms are [tex]\(y^3\)[/tex] and [tex]\(y\)[/tex]. Similarly, when multiplying exponents with the same base, you add the exponents:
[tex]\[
y^3 \cdot y = y^{3+1} = y^4
\][/tex]
Putting it all together, we get:
[tex]\[
(4 x^3 y^3) (2 x^2 y) = 8 x^5 y^4
\][/tex]
Thus, the simplified expression is:
[tex]\[
8 x^5 y^4
\][/tex]
The correct answer is [tex]\(\boxed{8 x^5 y^4}\)[/tex].
