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The given problem is to multiply and simplify the following expression:

\[
\frac{8a^3 b}{b^5} \cdot \frac{5b}{4a^2}
\]

### Step-by-Step Solution:

1. **Combine the two fractions**:
   Multiply the numerators and denominators:
   \[
   \frac{8a^3 b}{b^5} \cdot \frac{5b}{4a^2} = \frac{(8a^3 b)(5b)}{(b^5)(4a^2)}
   \]

   Simplify the numerator and denominator:
   \[
   \frac{(8 \cdot 5) a^3 b \cdot b}{(4a^2) b^5} = \frac{40a^3 b^2}{4a^2 b^5}
   \]

2. **Simplify the coefficients**:
   Divide \(40\) by \(4\):
   \[
   \frac{40a^3 b^2}{4a^2 b^5} = \frac{10a^3 b^2}{a^2 b^5}
   \]

3. **Simplify the variables**:
   - For \(a^3 / a^2\), subtract the exponents: \(a^{3-2} = a^1 = a\).
   - For \(b^2 / b^5\), subtract the exponents: \(b^{2-5} = b^{-3} = \frac{1}{b^3}\).

   The simplified expression becomes:
   \[
   \frac{10a}{b^3}
   \]

### Final Answer:
\[
\boxed{\frac{10a}{b^3}}
\]

Let me know if you'd like further clarification!

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Here are 5 related questions to expand your understanding:
1. How do you handle negative exponents in simplification?
2. What happens if one term in the numerator or denominator equals zero?
3. How do you multiply fractions with multiple variables?
4. What are some tips for identifying like terms when simplifying expressions?
5. How do you simplify expressions that involve roots instead of exponents?

**Tip**: When simplifying fractions, always handle coefficients and variables separately to avoid confusion.

Multiply and simplify the expression (8a^3b / b^5) * (5b / 4a^2).

Learn how to multiply and simplify the algebraic expression (8a^3b / b^5) * (5b / 4a^2) with a step-by-step solution. This problem covers exponent rules and fraction simplification, suitable for Grades 8-10.

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