Sure! Let's simplify the given expression step by step:
The given expression is:
[tex]\[
\left(4 v^3 w\right)\left(-2 w^3\right)^2
\][/tex]
### Step 1: Simplify [tex]\(\left(-2 w^3\right)^2\)[/tex]
First, we will deal with the term [tex]\(\left(-2 w^3\right)^2\)[/tex].
[tex]\[
\left(-2 w^3\right)^2 = \left(-2\right)^2 \cdot \left(w^3\right)^2
\][/tex]
Calculate each part separately:
- [tex]\(\left(-2\right)^2 = 4\)[/tex]
- [tex]\(\left(w^3\right)^2 = w^{3 \cdot 2} = w^6\)[/tex]
Therefore:
[tex]\[
\left(-2 w^3\right)^2 = 4 w^6
\][/tex]
### Step 2: Combine the results with [tex]\(4 v^3 w\)[/tex]
Next, multiply [tex]\(4 v^3 w\)[/tex] by the result from step 1.
[tex]\[
\left(4 v^3 w\right)\left(4 w^6\right)
\][/tex]
### Step 3: Simplify the combined expression
Now, combine the coefficients (numerical values) and the exponents of similar variables.
[tex]\[
4 \cdot 4 \cdot v^3 \cdot w \cdot w^6
\][/tex]
- Combine the coefficients: [tex]\(4 \cdot 4 = 16\)[/tex]
- Combine the [tex]\(w\)[/tex] terms using the laws of exponents: [tex]\(w^1 \cdot w^6 = w^{1+6} = w^7\)[/tex]
So, the expression simplifies to:
[tex]\[
16 v^3 w^7
\][/tex]
### Final Answer
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]
