The expression (a^(-3)b^(1/2))/(a^(-1/2)b^(-1)), where a gt 1 and b gt 1, is equivalent to which of the following?
GMAT Advanced Math : (Adv_Math) Questions
The expression \(\frac{\mathrm{a}^{-3}\mathrm{b}^{1/2}}{\mathrm{a}^{-1/2}\mathrm{b}^{-1}}\), where \(\mathrm{a} \gt 1\) and \(\mathrm{b} \gt 1\), is equivalent to which of the following?
1. TRANSLATE the problem information
- Given expression: \(\mathrm{a^{-3}b^{1/2} / a^{-1/2}b^{-1}}\)
- Need to simplify and match to one of the radical form answer choices
2. SIMPLIFY using the quotient rule for exponents
- Apply \(\mathrm{a^m / a^n = a^{m-n}}\) to each variable separately
- For a terms: \(\mathrm{a^{-3} / a^{-1/2} = a^{-3-(-1/2)} = a^{-3+1/2} = a^{-5/2}}\)
- For b terms: \(\mathrm{b^{1/2} / b^{-1} = b^{1/2-(-1)} = b^{1/2+1} = b^{3/2}}\)
3. SIMPLIFY the overall expression
- Combine results: \(\mathrm{a^{-5/2} \cdot b^{3/2}}\)
- Rewrite using positive exponents: \(\mathrm{b^{3/2} / a^{5/2}}\)
4. TRANSLATE to radical form
- Convert \(\mathrm{b^{3/2}}\): \(\mathrm{b^{3/2} = b^{1+1/2} = b^1 \cdot b^{1/2} = b\sqrt{b}}\)
- Convert \(\mathrm{a^{5/2}}\): \(\mathrm{a^{5/2} = a^{2+1/2} = a^2 \cdot a^{1/2} = a^2\sqrt{a}}\)
- Final form: \(\mathrm{b\sqrt{b} / (a^2\sqrt{a})}\)
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make sign errors when handling negative exponents in the quotient rule.
They might compute \(\mathrm{a^{-3} / a^{-1/2}}\) as \(\mathrm{a^{-3-1/2} = a^{-7/2}}\) instead of the correct \(\mathrm{a^{-3+1/2} = a^{-5/2}}\), forgetting that subtracting a negative number means adding. This leads to the wrong final exponent and an answer that doesn't match any of the choices, causing them to guess.
Second Most Common Error:
Poor TRANSLATE reasoning: Students struggle with the conversion from mixed fractional exponents to radical form.
They might leave the answer as \(\mathrm{b^{3/2} / a^{5/2}}\) without converting to radicals, or incorrectly convert \(\mathrm{b^{3/2}}\) to \(\mathrm{\sqrt{b^3}}\) instead of \(\mathrm{b\sqrt{b}}\). This prevents them from recognizing that their work matches Choice C.
The Bottom Line:
This problem tests whether students can carefully track signs and fractions when applying exponent rules, then fluently convert between exponential and radical notation. The key insight is that "subtracting a negative" means "adding a positive" - a concept that trips up many students in exponent arithmetic.