Simplify the algebraic expression (b^3 x^2 a^4 z^3) · (b^4 x^3 a^3 z^2) ÷ (a^2 b^4 z^3) and express the result in terms of a, b, x, and z.

Introduction / Context:
This question assesses your skill in manipulating algebraic expressions with multiple variables and exponents. Simplifying products and quotients of powers is a fundamental algebra technique used in higher mathematics, physics, and engineering formulas.

Given Data / Assumptions:

• Expression: (b^3 x^2 a^4 z^3) · (b^4 x^3 a^3 z^2) ÷ (a^2 b^4 z^3).
• Variables a, b, x, and z are non zero.
• We must simplify and express the final answer as a power of each variable.

Concept / Approach:
Key exponent rules:

• When multiplying like bases: a^m · a^n = a^(m + n).
• When dividing like bases: a^m ÷ a^n = a^(m − n).
• We can rearrange factors because multiplication is commutative and associative.

Step-by-Step Solution:
Consider the numerator: (b^3 x^2 a^4 z^3) · (b^4 x^3 a^3 z^2). Combine exponents for each variable: b^(3 + 4) x^(2 + 3) a^(4 + 3) z^(3 + 2). This gives b^7 x^5 a^7 z^5. Now divide by the denominator a^2 b^4 z^3. Apply exponent subtraction: a^(7 − 2) = a^5, b^(7 − 4) = b^3, z^(5 − 3) = z^2, and x^5 remains unchanged. Therefore the simplified expression is a^5 b^3 x^5 z^2, which can be written as b^3 x^5 a^5 z^2.

Verification / Alternative check:
As a numerical check, assign values a = 2, b = 3, x = 4, and z = 5. Compute the original expression and the simplified form with a calculator or carefully by hand. Both evaluations will yield the same numerical result, confirming that the algebraic simplification is correct.

Why Other Options Are Wrong:
Option a: b^2 x^4 a^6 z has incorrect exponents for b, x, a, and z compared to the correct subtraction and addition of exponents.
Option b: b^3 x^2 a^4 z^3 reproduces part of the original numerator and fails to include the effect of the second factor and the denominator.
Option d: b a z x severely underestimates the powers and ignores the exponent arithmetic.
Option e: a^7 b^3 x^5 z^2 has the wrong exponent for a since 7 − 2 should give 5, not 7.

Common Pitfalls:
Students sometimes add instead of subtracting exponents during division, or forget to combine like bases across both numerators before dividing. Another mistake is misreading variables and mixing up exponents for x and z. Carefully grouping all like terms before applying exponent rules helps avoid these issues.

Final Answer:
The simplified expression is b^3 x^5 a^5 z^2.