To simplify the given expression:

$\frac{12a^4b^5}{2a^2b}$

Step-by-step solution:

1. Simplify the coefficients: $\frac{12}{2} = 6$

2. Simplify the powers of $a$ using the rule $\frac{a^m}{a^n} = a^{m-n}$: $\frac{a^4}{a^2} = a^{4-2} = a^2$

3. Simplify the powers of $b$ using the same rule: $\frac{b^5}{b} = b^{5-1} = b^4$

4. Combine the simplified terms: $6a^2b^4$

Final Answer:

$6a^2b^4$

Do you have any questions or want further details?

Here are 5 related questions you might explore:

1. How does the rule of exponents apply when multiplying or dividing powers with the same base?
2. What happens if one of the variables in the numerator or denominator has a zero exponent?
3. How do you simplify expressions with negative exponents in a fraction?
4. Can the same approach be used if the bases are different (e.g., $a^4b^5 \div c^2$)?
5. How would you rewrite the result if you needed it in fractional form?

Tip: When simplifying expressions, always simplify coefficients, variables, and powers separately to avoid errors.