Certainly! Let's solve the given expression step-by-step.
Given the expression:
[tex]\[
\frac{uv}{3u - 6v} \times \frac{uU - 8v}{u^2v}
\][/tex]
### Step 1: Express Each Fraction
First, let's express the two fractions clearly:
1. [tex]\(\frac{uv}{3u - 6v}\)[/tex]
2. [tex]\(\frac{uU - 8v}{u^2v}\)[/tex]
### Step 2: Multiply the Fractions
When multiplying the fractions, we multiply the numerators together and the denominators together:
[tex]\[
\frac{uv}{3u - 6v} \times \frac{uU - 8v}{u^2v} = \frac{(uv) \cdot (uU - 8v)}{(3u - 6v) \cdot (u^2v)}
\][/tex]
### Step 3: Simplify the Numerator
First, simplify the numerator:
[tex]\[
(uv) \cdot (uU - 8v) = uv(uU - 8v) = uvuU - 8uvv = u^2Uv - 8uv^2
\][/tex]
### Step 4: Simplify the Denominator
Now, simplify the denominator:
[tex]\[
(3u - 6v) \cdot (u^2v) = u^2v(3u - 6v)
\][/tex]
### Step 5: Combine the Results
Putting it all together:
[tex]\[
\frac{u^2Uv - 8uv^2}{u^2v(3u - 6v)}
\][/tex]
### Step 6: Factor and Further Simplify
Factor out common terms if any:
[tex]\[
u^2Uv - 8uv^2 = u(uUv - 8v^2)
\][/tex]
Thus, we get:
[tex]\[
\frac{u(uUv - 8v^2)}{u^2v(3u - 6v)}
\][/tex]
We can cancel the common factor [tex]\(u\)[/tex] in the numerator and denominator:
[tex]\[
\frac{uUv - 8v^2}{uv(3u - 6v)}
\][/tex]
Again, cancel the common factor [tex]\(v\)[/tex] in the numerator and denominator:
[tex]\[
\frac{uU - 8v}{u(3u - 6v)}
\][/tex]
### Step 7: Further Simplify the Denominator
Notice that [tex]\(3u - 6v\)[/tex] can be factored:
[tex]\[
3u - 6v = 3(u - 2v)
\][/tex]
So, the expression simplifies to:
[tex]\[
\frac{uU - 8v}{u \cdot 3(u - 2v)}
\][/tex]
Finally, simplify the denominator:
[tex]\[
u \cdot 3(u - 2v) = 3u(u - 2v)
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{uU - 8v}{3u(u - 2v)}
\][/tex]
### Summary
So, step-by-step, we have simplified:
[tex]\[
\frac{uv}{3u - 6v} \times \frac{uU - 8v}{u^2v}
\][/tex]
to:
[tex]\[
\frac{uU - 8v}{3u(u - 2v)}
\][/tex]
