Let's simplify the expression:

\[
\frac{(5ab^2)^3}{25a^3b^2}
\]

### Step-by-step:

1. **Expand the numerator:**

   \[
   (5ab^2)^3 = 5^3 \cdot (a)^3 \cdot (b^2)^3
   \]
   Which gives:
   \[
   5^3 = 125, \quad a^3, \quad b^{6} \quad \text{(since } (b^2)^3 = b^{6}\text{)}
   \]
   Therefore, the numerator is:
   \[
   125a^3b^6
   \]

2. **The denominator remains as:**

   \[
   25a^3b^2
   \]

3. **Now, write the full expression:**

   \[
   \frac{125a^3b^6}{25a^3b^2}
   \]

4. **Simplify the constants:**

   \[
   \frac{125}{25} = 5
   \]

5. **Simplify the powers of \(a\):**

   Since both the numerator and denominator have \(a^3\), they cancel out:

   \[
   \frac{a^3}{a^3} = 1
   \]

6. **Simplify the powers of \(b\):**

   Using the rule of exponents \(\frac{b^6}{b^2} = b^{6-2} = b^4\):

   \[
   b^4
   \]

So the final simplified expression is:

\[
5b^4
\]

---

Would you like more details or have any questions?

Here are 5 related questions to further your understanding:
1. How do you simplify expressions involving multiple variables?
2. What are the rules for simplifying exponents in division?
3. How would you handle negative exponents in a similar problem?
4. What is the importance of understanding exponent rules in algebra?
5. Can you explain the general process for expanding a binomial raised to a power?

**Tip:** Always simplify constants first when working with fractions; it makes the process easier!

Learn how to simplify the expression ((5ab^2)^3) / (25a^3b^2) step by step. This problem demonstrates key algebraic techniques including exponent rules, fraction simplification, and working with variables. Suitable for students in Grades 9-10.

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