Question:Which expression is equivalent to (x^7 y^4 z^(-3))/(x^2 y^5 z^(-1)), where x, y, and z are positive?x^9 y^9 z^(-4)x^5 y^(-1)...
GMAT Advanced Math : (Adv_Math) Questions
Which expression is equivalent to \(\frac{\mathrm{x}^7 \mathrm{y}^4 \mathrm{z}^{-3}}{\mathrm{x}^2 \mathrm{y}^5 \mathrm{z}^{-1}}\), where x, y, and z are positive?
- \(\mathrm{x}^9 \mathrm{y}^9 \mathrm{z}^{-4}\)
- \(\mathrm{x}^5 \mathrm{y}^{-1} \mathrm{z}^{-2}\)
- \(\mathrm{x}^5 \mathrm{y}^{-1} \mathrm{z}^2\)
- \(\mathrm{x}^{-5} \mathrm{y} \mathrm{z}^2\)
1. INFER the approach needed
- This is a division problem with exponential expressions
- Since we have the same variables in numerator and denominator, we can use the quotient rule
- Strategy: Apply \(\mathrm{a^m / a^n = a^{(m-n)}}\) to each variable separately
2. SIMPLIFY each variable using the quotient rule
For the x terms:
- \(\mathrm{x^7 / x^2 = x^{(7-2)} = x^5}\)
For the y terms:
- \(\mathrm{y^4 / y^5 = y^{(4-5)} = y^{(-1)}}\)
For the z terms:
- \(\mathrm{z^{(-3)} / z^{(-1)} = z^{(-3-(-1))} = z^{(-3+1)} = z^{(-2)}}\)
3. SIMPLIFY by combining results
- Put together: \(\mathrm{x^5 y^{(-1)} z^{(-2)}}\)
- This matches choice (B) exactly
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY skill: Students make sign errors when working with negative exponents, particularly in the subtraction \(\mathrm{z^{(-3)} - (-1)}\).
Many students incorrectly compute \(\mathrm{z^{(-3)} - (-1)}\) as \(\mathrm{z^{(-3-1)} = z^{(-4)}}\) instead of \(\mathrm{z^{(-3+1)} = z^{(-2)}}\). They forget that subtracting a negative number means adding the positive. This leads them to select Choice A (\(\mathrm{x^9 y^9 z^{-4}}\)) after making additional errors, or causes confusion that leads to guessing.
Second Most Common Error:
Inadequate SIMPLIFY execution: Students add exponents instead of subtracting them, confusing division rules with multiplication rules.
They might compute \(\mathrm{x^7 / x^2}\) as \(\mathrm{x^{(7+2)} = x^9}\) and \(\mathrm{y^4 / y^5}\) as \(\mathrm{y^{(4+5)} = y^9}\), mixing up the quotient rule with the product rule. This systematic error leads them toward Choice A (\(\mathrm{x^9 y^9 z^{-4}}\)).
The Bottom Line:
This problem tests careful application of the quotient rule, especially with negative exponents. Success depends on methodical subtraction and attention to sign changes when subtracting negative numbers.