When the algebraic expression 256a^2 b^2 c^2 is divided by 64a^2, what is the resulting simplified expression in terms of a, b, and c?

Introduction / Context:This question tests basic manipulation of algebraic expressions involving powers and coefficients. It is essentially a division of one monomial by another. The goal is to simplify by dividing coefficients and subtracting exponents for matching variables, a common skill required in algebra and polynomial operations.

Given Data / Assumptions:

• The numerator is 256a^2 b^2 c^2.
• The denominator is 64a^2.
• All variables a, b, and c are assumed non zero for division to make sense.
• We are asked to write the simplified result in standard algebraic form.

Concept / Approach:When dividing monomials, divide the numerical coefficients and subtract the exponents of like variables. For example, a^m / a^n = a^(m − n) when a is not zero. Here the coefficient 256 is divided by 64, and a^2 in numerator is divided by a^2 in denominator. The remaining variables b^2 and c^2 stay as they are since there are no matching powers of b or c in the denominator.

Step-by-Step Solution:Write the expression as (256a^2 b^2 c^2) / (64a^2).First divide the coefficients: 256 / 64 = 4.Next divide the powers of a: a^2 / a^2 = a^(2 − 2) = a^0 = 1.The variables b^2 and c^2 remain unchanged because there are no b or c factors in the denominator.Thus the result is 4 * b^2 * c^2.So the simplified expression is 4b^2c^2.

Verification / Alternative check:To verify, choose simple non zero values for a, b, and c, such as a = 1, b = 2, c = 3. Compute the original fraction: numerator = 256 * 1^2 * 2^2 * 3^2 = 256 * 4 * 9 = 9216. Denominator = 64 * 1^2 = 64. Then 9216 / 64 = 144. Now compute 4b^2c^2 directly: 4 * 2^2 * 3^2 = 4 * 4 * 9 = 144. The results match, confirming the simplification is correct.

Why Other Options Are Wrong:

• 2bc^2 and 2b^2c result from dividing the coefficient incorrectly by a factor of 2 instead of 4.
• 4bc drops one power of both b and c, ignoring the original exponents b^2 and c^2.
• b^2c^2 misses the coefficient 4 entirely.
• Only 4b^2c^2 correctly accounts for both coefficients and exponents.

Common Pitfalls:

• Forgetting that a^2 / a^2 equals 1 and mistakenly leaving a factor of a in the final result.
• Dividing 256 by 64 incorrectly, especially if doing mental arithmetic under time pressure.
• Misreading the original expression and changing exponents of b or c when no division affects them.

Final Answer:4b^2c^2