Let's break down the expression step by step to simplify it:
Given expression:
[tex]\[
(4v^3w)(-2w^3)^2
\][/tex]
1. Simplify the inner part \((-2w^3)^2\):
- When you raise a product to a power, you raise each factor in the product to that power. Hence:
[tex]\[
(-2w^3)^2 = (-2)^2 \cdot (w^3)^2
\][/tex]
- Calculate \((-2)^2\):
[tex]\[
(-2)^2 = 4
\][/tex]
- Calculate \((w^3)^2\):
[tex]\[
(w^3)^2 = w^{3 \cdot 2} = w^6
\][/tex]
- Putting it all together, we get:
[tex]\[
(-2w^3)^2 = 4w^6
\][/tex]
2. Substitute back into the original expression:
[tex]\[
(4v^3w)(4w^6)
\][/tex]
3. Multiply the coefficients and combine like terms:
- Multiply the numerical coefficients:
[tex]\[
4 \cdot 4 = 16
\][/tex]
- Combine the \(v\) terms:
[tex]\[
v^3
\][/tex]
Since there is only one \(v\) term, it remains \(v^3\).
- Combine the \(w\) terms:
[tex]\[
w \cdot w^6 = w^{1+6} = w^7
\][/tex]
Putting all the parts together, we get the simplified expression:
[tex]\[
16v^3w^7
\][/tex]
So,
[tex]\[
(4v^3w)(-2w^3)^2 = 16v^3w^7
\][/tex]
