To multiply the given rational expressions, let's follow these steps carefully.
### Given Expressions:
[tex]\[
\frac{-3 u^2 w}{7 v^3 w^2} \quad \text{and} \quad \frac{49 u v^2}{w^3}
\][/tex]
### Step 1: Multiply the Numerators
First, multiply the numerators of the two fractions:
[tex]\[
-3 u^2 w \times 49 u v^2
\][/tex]
### Step 2: Multiply the Denominators
Next, multiply the denominators of the two fractions:
[tex]\[
7 v^3 w^2 \times w^3
\][/tex]
### Step 3: Perform the Multiplications
#### Numerator:
[tex]\[
-3 \times 49 = -147
\][/tex]
[tex]\[
u^2 \times u = u^{2+1} = u^3
\][/tex]
[tex]\[
w \times v^2 = v^2 w
\][/tex]
So, the combined numerator is:
[tex]\[
-147 u^3 v^2 w
\][/tex]
#### Denominator:
[tex]\[
7 \times 1 = 7
\][/tex]
[tex]\[
v^3 \times v^0 = v^3
\][/tex]
[tex]\[
w^2 \times w^3 = w^{2+3} = w^5
\][/tex]
So, the combined denominator is:
[tex]\[
7 v^3 w^5
\][/tex]
### Step 4: Combine the Numerators and Denominators
Now, place the combined numerator over the combined denominator:
[tex]\[
\frac{-147 u^3 v^2 w}{7 v^3 w^5}
\][/tex]
### Step 5: Simplify the Expression
Combine like terms to simplify the expression:
For the constants:
[tex]\[
\frac{-147}{7} = -21
\][/tex]
For the variable [tex]\( u \)[/tex]:
[tex]\[
u^3 \quad (\text{no change needed})
\][/tex]
For the variable [tex]\( v \)[/tex]:
[tex]\[
v^{2-3} = v^{-1}
\][/tex]
For the variable [tex]\( w \)[/tex]:
[tex]\[
w^{1-5} = w^{-4}
\][/tex]
Thus, the simplified rational expression is:
[tex]\[
-21 u^3 v^{-1} w^{-4}
\][/tex]
### Final Answer
Therefore, the product of the given rational expressions is:
[tex]\[
\boxed{-21 u^3 v^{-1} w^{-4}}
\][/tex]
