Question:Which expression is equivalent to sqrt[5]{x^8/y^8}, where x and y are positive?\((\frac{\mathrm{x}}{\mathrm{y}})^{\frac{5}{8}}\)\((\frac{\mat...
GMAT Advanced Math : (Adv_Math) Questions
Which expression is equivalent to \(\sqrt[5]{\frac{\mathrm{x}^8}{\mathrm{y}^8}}\), where x and y are positive?
- \((\frac{\mathrm{x}}{\mathrm{y}})^{\frac{5}{8}}\)
- \((\frac{\mathrm{x}}{\mathrm{y}})^{\frac{8}{5}}\)
- \((\frac{\mathrm{x}}{\mathrm{y}})^3\)
- \((\frac{\mathrm{x}}{\mathrm{y}})^{13}\)
1. TRANSLATE the radical notation
- Given: Fifth root of \(\mathrm{x^8/y^8}\)
- TRANSLATE this to exponential form: \(\mathrm{(x^8/y^8)^{1/5}}\)
- Key insight: The fifth root means raising to the \(\mathrm{1/5}\) power
2. INFER the solution strategy
- We need to simplify this complex exponential expression
- Strategy: Use power rules to distribute and multiply exponents
- Work from outside to inside: handle the \(\mathrm{1/5}\) power first
3. SIMPLIFY using exponent distribution
- Distribute the \(\mathrm{1/5}\) power to both numerator and denominator:
\(\mathrm{(x^8/y^8)^{1/5} = (x^8)^{1/5} / (y^8)^{1/5}}\)
4. SIMPLIFY using the power rule
- Apply \(\mathrm{(a^m)^n = a^{mn}}\) to each term:
- \(\mathrm{(x^8)^{1/5} = x^{8×1/5} = x^{8/5}}\)
- \(\mathrm{(y^8)^{1/5} = y^{8×1/5} = y^{8/5}}\)
5. SIMPLIFY the final expression
- Combine: \(\mathrm{x^{8/5} / y^{8/5} = (x/y)^{8/5}}\)
Answer: B. \(\mathrm{(x/y)^{8/5}}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students might confuse which number goes in the numerator vs denominator of the fractional exponent.
They might think "fifth root" means the 5 should be in the numerator, writing \(\mathrm{(x^8/y^8)^{5/8}}\) instead of \(\mathrm{(x^8/y^8)^{1/5}}\). This reasoning flips the fraction and leads them through correct algebraic steps to get \(\mathrm{(x/y)^{5/8}}\).
This may lead them to select Choice A (\(\mathrm{(x/y)^{5/8}}\)).
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly set up \(\mathrm{(x^8/y^8)^{1/5}}\) but make arithmetic errors when multiplying \(\mathrm{8 × 1/5}\).
They might calculate \(\mathrm{8 × 1/5 = 5/8}\) instead of \(\mathrm{8/5}\), confusing the order. This leads them to \(\mathrm{x^{5/8}/y^{5/8} = (x/y)^{5/8}}\).
This may lead them to select Choice A (\(\mathrm{(x/y)^{5/8}}\)).
The Bottom Line:
The key challenge is accurately translating radical notation into fractional exponents, then carefully handling the arithmetic when multiplying fractions. Most errors stem from confusion about which number (5 or 8) belongs in the numerator of the final exponent.