In basic algebraic simplification, when the product expression 24a^2b^2 is divided by 6b^2, what is the resulting simplified expression written only in terms of the variable a?

Introduction / Context:
This question tests basic algebraic simplification involving powers of variables and cancellation of common factors. Such problems are common in school algebra and aptitude exams and help you practice how to handle products of powers when one expression is divided by another. The goal is to recognise which factors cancel out completely and which powers remain in the final simplified result.

Given Data / Assumptions:

• The original algebraic expression is 24a^2b^2.
• This expression is divided by 6b^2.
• a and b are real variables.

Concept / Approach:
The key ideas are:

• Division of coefficients is done as ordinary number division.
• For variables with powers, when dividing like bases, subtract the exponents: a^m / a^n = a^(m − n).
• If the power becomes zero, the variable factor becomes 1.
• Any common factor in numerator and denominator can be cancelled.

Step-by-Step Solution:
Start with the expression: 24a^2b^2 ÷ 6b^2. Rewrite it as a single fraction: (24a^2b^2) / (6b^2). Simplify the numerical coefficients: 24 / 6 = 4. Simplify the b terms using exponent rules: b^2 / b^2 = b^(2−2) = b^0 = 1. The a term is only in the numerator: a^2 remains unchanged. Therefore the simplified expression is 4a^2.

Verification / Alternative check:
We can verify by substituting simple numeric values for a and b. For example, let a = 2 and b = 3. Then 24a^2b^2 = 24 * 4 * 9 = 864. The denominator 6b^2 = 6 * 9 = 54. The fraction 864 / 54 = 16. Now evaluate 4a^2: 4 * 4 = 16. The values match, confirming that 4a^2 is correct.

Why Other Options Are Wrong:
Option b: 4ab keeps a factor of b, but b^2 cancels completely, so this is incorrect.
Option c: 4b also ignores the remaining a^2, so it does not match the simplification.
Option d: 4 drops both a^2 and b^2, losing the variable dependence.
Option e: a^2b^2 / 6 is an unsimplified rearrangement and does not reflect the required division by 6b^2.

Common Pitfalls:
A common mistake is to divide coefficients correctly but ignore powers of variables, or to cancel only one factor of b instead of both b^2 terms. Another typical error is to think that everything cancels and the answer becomes just the coefficient. Careful application of exponent rules prevents these errors.

Final Answer:
The simplified result of dividing 24a^2b^2 by 6b^2 is 4a^2.