Certainly! Let's divide the given algebraic expression step-by-step.
We need to simplify the expression:
[tex]\[
\frac{12 p^3 q - 16 p^5 q^2 + 10 p^4 q^4}{8 p^2 q^3}
\][/tex]
### Step 1: Separate the Expression
We can break down the numerator and divide each term by the denominator individually:
[tex]\[
\frac{12 p^3 q}{8 p^2 q^3} - \frac{16 p^5 q^2}{8 p^2 q^3} + \frac{10 p^4 q^4}{8 p^2 q^3}
\][/tex]
### Step 2: Simplify Each Term
1. Simplify [tex]\(\frac{12 p^3 q}{8 p^2 q^3}\)[/tex]:
[tex]\[
\frac{12 p^3 q}{8 p^2 q^3} = \frac{12}{8} \cdot \frac{p^3}{p^2} \cdot \frac{q}{q^3} = \frac{3}{2} \cdot p^{3-2} \cdot q^{1-3} = \frac{3}{2} \cdot p \cdot q^{-2} = \frac{3 p}{2 q^2}
\][/tex]
2. Simplify [tex]\(\frac{16 p^5 q^2}{8 p^2 q^3}\)[/tex]:
[tex]\[
\frac{16 p^5 q^2}{8 p^2 q^3} = \frac{16}{8} \cdot \frac{p^5}{p^2} \cdot \frac{q^2}{q^3} = 2 \cdot p^{5-2} \cdot q^{2-3} = 2 \cdot p^3 \cdot q^{-1} = \frac{2 p^3}{q}
\][/tex]
3. Simplify [tex]\(\frac{10 p^4 q^4}{8 p^2 q^3}\)[/tex]:
[tex]\[
\frac{10 p^4 q^4}{8 p^2 q^3} = \frac{10}{8} \cdot \frac{p^4}{p^2} \cdot \frac{q^4}{q^3} = \frac{5}{4} \cdot p^{4-2} \cdot q^{4-3} = \frac{5}{4} \cdot p^2 \cdot q = \frac{5 p^2 q}{4}
\][/tex]
### Step 3: Combine the Simplified Terms
Now, combine the simplified terms:
[tex]\[
\frac{3 p}{2 q^2} - \frac{2 p^3}{q} + \frac{5 p^2 q}{4}
\][/tex]
### Step 4: Find a Common Denominator
To combine these fractions, we'll find the common denominator. The common denominator here is [tex]\(4 q^2\)[/tex].
Rewrite the simplified terms with the common denominator [tex]\(4 q^2\)[/tex]:
[tex]\[
\frac{3 p}{2 q^2} = \frac{3 p \cdot 2}{2 q^2 \cdot 2} = \frac{6 p}{4 q^2}
\][/tex]
[tex]\[
-\frac{2 p^3}{q} = -\frac{2 p^3 \cdot 4 q}{q \cdot 4 q^2} = -\frac{8 p^3 q}{4 q^2}
\][/tex]
[tex]\[
\frac{5 p^2 q}{4} = \frac{5 p^2 q}{4 q^2} = \frac{5 p^2 q^3}{4 q^2}
\][/tex]
### Step 5: Combine the Fractions
Now that we have a common denominator, combine the fractions:
[tex]\[
\frac{6 p}{4 q^2} - \frac{8 p^3 q}{4 q^2} + \frac{5 p^2 q^3}{4 q^2}
\][/tex]
Combine the numerators over the common denominator:
[tex]\[
\frac{6 p - 8 p^3 q + 5 p^2 q^3}{4 q^2}
\][/tex]
### Step 6: Factor the Numerator
Notice that the numerator can be factored:
[tex]\[
\frac{6 p - 8 p^3 q + 5 p^2 q^3}{4 q^2} = \frac{p(6 - 8 p^2 q + 5 p q^3)}{4 q^2}
\][/tex]
Rewriting this in a clearer form gives us:
[tex]\[
\boxed{\frac{p(6 - 8 p^2 q + 5 p q^3)}{4 q^2}}
\][/tex]
So, the simplified expression is:
[tex]\[
\frac{p(-8 p^2 q + 5 p q^3 + 6)}{4 q^2}
\][/tex]
