Which expression is equivalent to \(\frac{(\mathrm{x}^5 \mathrm{y}^{-2})^3}{(\mathrm{x}^{-1} \mathrm{y}^4)^2}\), where x gt 0 and y gt 0? x^(17)/y^(...
GMAT Advanced Math : (Adv_Math) Questions
Which expression is equivalent to \(\frac{(\mathrm{x}^5 \mathrm{y}^{-2})^3}{(\mathrm{x}^{-1} \mathrm{y}^4)^2}\), where \(\mathrm{x} \gt 0\) and \(\mathrm{y} \gt 0\)?
- \(\frac{\mathrm{x}^{17}}{\mathrm{y}^{14}}\)
- \(\frac{\mathrm{x}^{13}}{\mathrm{y}^{14}}\)
- \(\mathrm{x}^{17} \mathrm{y}^{14}\)
- \(\frac{\mathrm{y}^{14}}{\mathrm{x}^{17}}\)
1. INFER the solution strategy
- We have a fraction with exponential expressions in both numerator and denominator, each raised to a power
- Strategy: Apply power rule first to eliminate outer parentheses, then use quotient rule to divide
2. SIMPLIFY by applying the power rule to each factor
- Numerator: \((\mathrm{x}^5 \mathrm{y}^{-2})^3\)
\(= \mathrm{x}^{5 \cdot 3} \cdot \mathrm{y}^{(-2) \cdot 3}\)
\(= \mathrm{x}^{15} \cdot \mathrm{y}^{-6}\)
- Denominator: \((\mathrm{x}^{-1} \mathrm{y}^4)^2\)
\(= \mathrm{x}^{(-1) \cdot 2} \cdot \mathrm{y}^{4 \cdot 2}\)
\(= \mathrm{x}^{-2} \cdot \mathrm{y}^8\)
3. SIMPLIFY by applying the quotient rule
- For x terms: \(\mathrm{x}^{15} / \mathrm{x}^{-2}\)
\(= \mathrm{x}^{15-(-2)}\)
\(= \mathrm{x}^{15+2}\)
\(= \mathrm{x}^{17}\)
- For y terms: \(\mathrm{y}^{-6} / \mathrm{y}^8\)
\(= \mathrm{y}^{(-6)-8}\)
\(= \mathrm{y}^{-14}\)
4. SIMPLIFY the final expression
\(\mathrm{x}^{17} \cdot \mathrm{y}^{-14}\)
\(= \mathrm{x}^{17} \cdot (1/\mathrm{y}^{14})\)
\(= \mathrm{x}^{17}/\mathrm{y}^{14}\)
Answer: A. \(\mathrm{x}^{17}/\mathrm{y}^{14}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution with negative exponents: Students make sign errors when subtracting negative exponents, particularly calculating \(15-(-2)\) as 13 instead of 17.
They might get: \(\mathrm{x}^{15-(-2)} = \mathrm{x}^{13}\), leading to \(\mathrm{x}^{13}/\mathrm{y}^{14}\).
This may lead them to select Choice B (\(\mathrm{x}^{13}/\mathrm{y}^{14}\))
Second Most Common Error:
Poor INFER reasoning about order of operations: Students attempt to divide before fully applying the power rule, creating confusion about which exponent rules to apply when.
This leads to getting stuck partway through and making calculation errors or abandoning systematic solution and guessing.
The Bottom Line:
This problem requires careful attention to signs when working with negative exponents. The key insight is recognizing that subtracting a negative exponent means adding the absolute value: \(15-(-2) = 15+2 = 17\).