When 8a^2 b^2 c^2 is divided by 4a^2, what is the simplified algebraic expression for the quotient?

Introduction / Context:
This question checks your comfort with basic algebraic simplification involving powers of variables. Expressions with exponents appear frequently in aptitude tests, and being able to divide and cancel common factors quickly will save valuable time in the exam.

Given Data / Assumptions:

- Dividend: 8a^2 b^2 c^2
- Divisor: 4a^2
- All variables a, b, and c are assumed to be non zero real numbers
- We are asked to find the simplified quotient

Concept / Approach:
When dividing algebraic terms with the same base, we subtract the exponents. Also, when dividing coefficients, we treat them as normal numbers. Thus, 8a^2 b^2 c^2 divided by 4a^2 can be handled by splitting the coefficient part and the variable part and simplifying each step by step.

Step-by-Step Solution:
Step 1: Write the division explicitly as (8a^2 b^2 c^2) / (4a^2).Step 2: Separate coefficients and variables: (8 / 4) * (a^2 / a^2) * b^2 * c^2.Step 3: Simplify the coefficients: 8 / 4 = 2.Step 4: Simplify the a terms. Since a^2 / a^2 = a^(2 − 2) = a^0 = 1, the factor in a disappears.Step 5: The remaining expression is 2 * b^2 * c^2, which we write as 2b^2c^2.

Verification / Alternative check:
You can check the result by substituting simple non zero integer values for a, b, and c. For example, choose a = 1, b = 2, c = 3. The original expression gives (8 * 1^2 * 2^2 * 3^2) / (4 * 1^2) = (8 * 4 * 9) / 4 = 288 / 4 = 72. The simplified expression 2b^2c^2 gives 2 * 4 * 9 = 72, which matches exactly.

Why Other Options Are Wrong:
- 2b^2 ignores the c^2 factor and so is incomplete.
- 2c^2 ignores the b^2 factor and is also incomplete.
- 2bc^2 has only a single power of b instead of b^2, so the exponent is incorrect.
- b^2c^2 misses the factor 2, so the numerical coefficient is wrong.

Common Pitfalls:
Common mistakes include forgetting to simplify the coefficients, mishandling exponents when dividing like bases, or accidentally cancelling extra powers of a variable. Always check that the power of each variable in the result is obtained by subtracting exponents correctly, and ensure that the coefficient is simplified as a normal fraction.

Final Answer:
The simplified quotient is 2b^2c^2.