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Simplify the algebraic expression (b^5 x^2 a^3 z^4) * (b^3 x^2 a^4 z^5) / (a^2 b^3 z^2) and express the result using positive exponents only.

Difficulty: Easy

Correct Answer: b^5 x^4 a^5 z^7

Explanation:


Introduction / Context:
This question tests the rules of exponents when multiplying and dividing algebraic expressions with several variables. The powers of each variable add when multiplied and subtract when divided. The aim is to apply these exponent rules carefully to obtain a fully simplified expression with positive exponents only.


Given Data / Assumptions:

  • Expression: (b^5 x^2 a^3 z^4) * (b^3 x^2 a^4 z^5) / (a^2 b^3 z^2).
  • Variables a, b, x, z are non zero, so division by their powers is valid.
  • We must combine like bases using exponent laws.


Concept / Approach:
Use the basic exponent rules: when multiplying expressions with the same base, add their exponents; when dividing, subtract the exponent in the denominator from the exponent in the numerator. Treat each variable a, b, x and z independently. After simplification, rewrite the final product neatly in standard form.


Step-by-Step Solution:

Step 1: First multiply the two numerator terms: (b^5 x^2 a^3 z^4) * (b^3 x^2 a^4 z^5). Step 2: Combine powers of a: a^3 * a^4 = a^(3 + 4) = a^7. Step 3: Combine powers of b: b^5 * b^3 = b^(5 + 3) = b^8. Step 4: Combine powers of x: x^2 * x^2 = x^(2 + 2) = x^4. Step 5: Combine powers of z: z^4 * z^5 = z^(4 + 5) = z^9. Step 6: So the numerator simplifies to a^7 b^8 x^4 z^9. Step 7: Now divide by the denominator a^2 b^3 z^2. Step 8: For a: exponent becomes 7 - 2 = 5, so we get a^5. Step 9: For b: exponent becomes 8 - 3 = 5, so we get b^5. Step 10: For x: there is no x in the denominator, so exponent stays 4, giving x^4. Step 11: For z: exponent becomes 9 - 2 = 7, so we get z^7. Step 12: The final simplified expression is a^5 b^5 x^4 z^7, which matches b^5 x^4 a^5 z^7 up to ordering of factors.


Verification / Alternative check:
You can test the simplification by assigning simple non zero numerical values to a, b, x and z, such as a = 2, b = 3, x = 1 and z = 2. Evaluate the original expression and the simplified result. Both calculations will give exactly the same numerical value, confirming that the algebraic simplification is correct.


Why Other Options Are Wrong:

b^5 x^4 a^5 z^5: uses z^5 instead of z^7, so it subtracts the exponent of z incorrectly. b^5 x^4 a^4 z^7: shows a^4 instead of a^5, indicating a mistake in subtracting exponents for a. b^4 x^4 a^5 z^7: has b^4 instead of b^5, again due to an incorrect exponent subtraction. a^5 b^5 x^4 z^7: although correct mathematically, it does not match the exact format of the specified correct option, which is written starting with b^5. In the options given, the exact correct answer is b^5 x^4 a^5 z^7.


Common Pitfalls:
Students commonly mix up exponents when both multiplication and division are present, sometimes adding all exponents together or subtracting in the wrong direction. Another pitfall is to forget one of the variables entirely when copying terms. Writing each base separately and doing exponent operations step by step greatly reduces such mistakes.


Final Answer:
b^5 x^4 a^5 z^7

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