To rewrite the given expression [tex]\( \left(4 v^3 w\right)\left(-2 w^3\right)^2 \)[/tex] in its simplest form, let's break it down into simpler components and simplify each step by step.
1. Analyze the Expression:
[tex]\[
\left(4 v^3 w\right)\left(-2 w^3\right)^2
\][/tex]
2. Evaluate the Power:
The expression [tex]\(\left(-2 w^3\right)^2\)[/tex] involves squaring [tex]\(-2 w^3\)[/tex].
[tex]\[
(-2 w^3)^2 = (-2)^2 (w^3)^2 = 4 w^6
\][/tex]
3. Substitute and Simplify:
Now, substitute the simplified part back into the original expression:
[tex]\[
\left(4 v^3 w\right) \times \left(4 w^6\right)
\][/tex]
4. Combine the Constants:
Multiply the constants [tex]\(4\)[/tex] and [tex]\(4\)[/tex]:
[tex]\[
4 \times 4 = 16
\][/tex]
5. Combine the Variables:
Multiply the variable parts:
[tex]\[
v^3 w \times w^6
\][/tex]
Using the laws of exponents, where [tex]\(w \times w^n = w^{1+n} = w^{1+6} = w^7\)[/tex], we get:
[tex]\[
v^3 w^7
\][/tex]
6. Combine Everything:
Put together the constant and variable parts:
[tex]\[
16 v^3 w^7
\][/tex]
So, the simplified form of the expression [tex]\( \left(4 v^3 w\right)\left(-2 w^3\right)^2 \)[/tex] is:
[tex]\[
16 v^3 w^7
\][/tex]
