Sure, let's work through reducing the given algebraic fraction step by step.
The expression to reduce is:
[tex]\[
\frac{a^3 + a^2 b}{5a} \cdot \frac{25}{3b + 3a}
\][/tex]
Step 1: Simplify each individual fraction.
For the first fraction:
[tex]\[
\frac{a^3 + a^2 b}{5a}
\][/tex]
Notice that we can factor [tex]\(a\)[/tex] out of the numerator:
[tex]\[
a^3 + a^2 b = a^2(a + b)
\][/tex]
So, the first fraction becomes:
[tex]\[
\frac{a^2(a + b)}{5a}
\][/tex]
We can simplify this by canceling [tex]\(a\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{a(a + b)}{5}
\][/tex]
For the second fraction:
[tex]\[
\frac{25}{3b + 3a}
\][/tex]
We can factor out a 3 from the denominator:
[tex]\[
3b + 3a = 3(b + a)
\][/tex]
So, the second fraction becomes:
[tex]\[
\frac{25}{3(b + a)}
\][/tex]
Step 2: Multiply the simplified fractions together:
[tex]\[
\frac{a(a + b)}{5} \cdot \frac{25}{3(b + a)}
\][/tex]
When multiplying fractions, we multiply the numerators together and the denominators together:
[tex]\[
\frac{a(a + b) \cdot 25}{5 \cdot 3(b + a)}
\][/tex]
Step 3: Simplify the resulting fraction.
Notice that [tex]\((a + b)\)[/tex] in the numerator and the denominator will cancel out:
[tex]\[
\frac{a \cdot 25}{5 \cdot 3}
\][/tex]
Also, 25 divided by 5 simplifies to 5:
[tex]\[
\frac{a \cdot 5}{3}
\][/tex]
So, the simplified expression is:
[tex]\[
\frac{5a}{3}
\][/tex]
Therefore, the reduced form of the given algebraic fraction is:
[tex]\[
\frac{5a}{3}
\][/tex]
