Sure, let's simplify the given expression step by step:
The expression given is:
[tex]\[
(4 v^3 w) \left( -2 w^3 \right)^2
\][/tex]
### Step 1: Simplify inside the parentheses
First, simplify the expression inside the parentheses:
[tex]\[
(-2 w^3)^2
\][/tex]
When squaring a product, you square each factor separately:
[tex]\[
(-2)^2 \cdot (w^3)^2
\][/tex]
Calculate [tex]\((-2)^2\)[/tex]:
[tex]\[
(-2)^2 = 4
\][/tex]
Calculate [tex]\((w^3)^2\)[/tex]:
[tex]\[
(w^3)^2 = w^{3 \cdot 2} = w^6
\][/tex]
So,
[tex]\[
(-2 w^3)^2 = 4 w^6
\][/tex]
### Step 2: Multiply the results with the remaining terms
Now, multiply the simplified expression with [tex]\(4 v^3 w\)[/tex]:
[tex]\[
(4 v^3 w) \cdot (4 w^6)
\][/tex]
### Step 3: Combine the constants
First, combine the constants:
[tex]\[
4 \cdot 4 = 16
\][/tex]
### Step 4: Combine the variables with the same base
Combine the [tex]\(w\)[/tex] terms:
[tex]\[
w \cdot w^6 = w^{1+6} = w^7
\][/tex]
### Step 5: Write the final expression
Now, multiply everything together:
[tex]\[
16 \cdot v^3 \cdot w^7 = 16 v^3 w^7
\][/tex]
So, the expression [tex]\((4 v^3 w) \left( -2 w^3 \right)^2\)[/tex] simplifies to:
[tex]\[
16 v^3 w^7
\][/tex]
