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:

$5b^4$

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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!