(x^(-2)y^(1/2))/(x^(1/3)y^(-1))The expression above, where x gt 1 and y gt 1, is equivalent to which of the following?
GMAT Advanced Math : (Adv_Math) Questions
\(\frac{\mathrm{x}^{-2}\mathrm{y}^{1/2}}{\mathrm{x}^{1/3}\mathrm{y}^{-1}}\)
The expression above, where \(\mathrm{x} \gt 1\) and \(\mathrm{y} \gt 1\), is equivalent to which of the following?
\(\frac{\sqrt{\mathrm{y}}}{\sqrt[3]{\mathrm{x}^2}}\)
\(\frac{\mathrm{y}\sqrt{\mathrm{y}}}{\sqrt[3]{\mathrm{x}^2}}\)
\(\frac{\mathrm{y}\sqrt{\mathrm{y}}}{\mathrm{x}\sqrt{\mathrm{x}}}\)
\(\frac{\mathrm{y}\sqrt{\mathrm{y}}}{\mathrm{x}^2\sqrt[3]{\mathrm{x}}}\)
1. TRANSLATE the division into separate fractions
- Given: \(\mathrm{x^{-2}y^{1/2} / x^{1/3}y^{-1}}\)
- Rewrite as: \(\mathrm{x^{-2}/x^{1/3} \cdot y^{1/2}/y^{-1}}\)
- This separates the x terms and y terms for easier handling
2. SIMPLIFY using exponent division rules
- For x terms: \(\mathrm{x^{-2}/x^{1/3} = x^{-2-1/3}}\)
- Calculate the exponent: \(\mathrm{-2 - 1/3 = -6/3 - 1/3 = -7/3}\)
- For y terms: \(\mathrm{y^{1/2}/y^{-1} = y^{1/2-(-1)} = y^{1/2+1} = y^{3/2}}\)
- Result: \(\mathrm{x^{-7/3} \cdot y^{3/2}}\)
3. TRANSLATE to radical form to match answer choices
- Convert \(\mathrm{x^{-7/3}}\): Since it's negative, move to denominator: \(\mathrm{1/x^{7/3}}\)
- Break down \(\mathrm{x^{7/3} = x^{6/3 + 1/3} = x^2 \cdot x^{1/3} = x^2\sqrt[3]{x}}\)
- Convert \(\mathrm{y^{3/2}}\): \(\mathrm{y^{3/2} = y^{1+1/2} = y^1 \cdot y^{1/2} = y\sqrt{y}}\)
- Final form: \(\mathrm{y\sqrt{y}/(x^2\sqrt[3]{x})}\)
Answer: D. \(\mathrm{y\sqrt{y}/(x^2\sqrt[3]{x})}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students incorrectly combine the fractional exponents, especially making arithmetic errors with \(\mathrm{-2 - 1/3}\). Many calculate this as \(\mathrm{-2 + 1/3 = -5/3}\) instead of \(\mathrm{-2 - 1/3 = -7/3}\).
This leads to \(\mathrm{x^{-5/3}y^{3/2}}\), which when converted to radical form gives something like \(\mathrm{y\sqrt{y}/(x\sqrt[3]{x^2})}\) instead of the correct \(\mathrm{y\sqrt{y}/(x^2\sqrt[3]{x})}\). This may lead them to select Choice A or B depending on their radical conversion.
Second Most Common Error:
Poor TRANSLATE reasoning: Students struggle with converting between exponential and radical notation, particularly with mixed exponents like \(\mathrm{x^{7/3}}\). They might write this as \(\mathrm{\sqrt[3]{x^7}}\) instead of properly decomposing it as \(\mathrm{x^2\sqrt[3]{x}}\).
This can cause confusion about which answer choice matches their work, leading to abandoning systematic solution and guessing.
The Bottom Line:
This problem tests precision with fractional exponent arithmetic and the ability to fluidly convert between exponential and radical forms. Small arithmetic errors in combining exponents cascade into wrong final answers.
\(\frac{\sqrt{\mathrm{y}}}{\sqrt[3]{\mathrm{x}^2}}\)
\(\frac{\mathrm{y}\sqrt{\mathrm{y}}}{\sqrt[3]{\mathrm{x}^2}}\)
\(\frac{\mathrm{y}\sqrt{\mathrm{y}}}{\mathrm{x}\sqrt{\mathrm{x}}}\)
\(\frac{\mathrm{y}\sqrt{\mathrm{y}}}{\mathrm{x}^2\sqrt[3]{\mathrm{x}}}\)