Simplify the algebraic expression [(b^2 x^3 a^2 z^4) * (b^2 x^4 a^4 z^3)] divided by (a^3 b^2 z^4). What is the resulting single term?

Introduction / Context:
This algebra question focuses on simplifying a product and quotient of monomials. Such questions test understanding of the laws of exponents and how to combine powers of the same variable when multiplying and dividing algebraic expressions. Mastery of these rules is important in many areas of mathematics, including polynomials and algebraic fractions.

Given Data / Assumptions:

• Expression: [(b^2 x^3 a^2 z^4) * (b^2 x^4 a^4 z^3)] / (a^3 b^2 z^4)
• a, b, x, and z are non zero real numbers or algebraic symbols.
• We need to express the result as a single simplified monomial.

Concept / Approach:
We use these exponent rules:

• When multiplying like bases: m^p * m^q = m^(p + q)
• When dividing like bases: m^p / m^q = m^(p - q)

We handle each variable separately, first combining exponents in the numerator, then subtracting the exponents from the denominator.

Step-by-Step Solution:
First monomial: b^2 x^3 a^2 z^4 Second monomial: b^2 x^4 a^4 z^3 Multiply them: for a, exponents are 2 and 4, giving a^(2 + 4) = a^6 For b, exponents are 2 and 2, giving b^(2 + 2) = b^4 For x, exponents are 3 and 4, giving x^(3 + 4) = x^7 For z, exponents are 4 and 3, giving z^(4 + 3) = z^7 So the numerator simplifies to a^6 b^4 x^7 z^7 Denominator is a^3 b^2 z^4 Divide powers: a^6 / a^3 = a^(6 - 3) = a^3 b^4 / b^2 = b^(4 - 2) = b^2 x^7 remains x^7, since there is no x in the denominator z^7 / z^4 = z^(7 - 4) = z^3 Final simplified expression: a^3 b^2 x^7 z^3

Verification / Alternative check:
We can verify by assigning simple values to the variables, for example a = 2, b = 2, x = 2, z = 2. Compute the original expression and the simplified expression numerically. Both should produce the same numerical value, confirming that the algebraic simplification is correct.

Why Other Options Are Wrong:
Option b uses b^3 instead of b^2, which would require an extra factor of b in the numerator. Option c has x^5 instead of x^7, as if two powers of x were lost. Option d has z^2 instead of z^3, suggesting a subtraction error in the exponent. Option e reduces the exponent of a to 2, which does not match the combination of a^6 over a^3.

Common Pitfalls:
Students often confuse when to add or subtract exponents. Exponents are added when multiplying like bases and subtracted when dividing. Another common mistake is to try to simplify coefficients and exponents together without separating variables. Writing out each exponent operation clearly for a, b, x, and z helps avoid these errors.

Final Answer:
The simplified single term is a^3 b^2 x^7 z^3.