Which expression is equivalent to \((\mathrm{xy})^{\frac{8}{3}}\), where x and y are positive?\(\sqrt[8]{(\mathrm{xy})^3}\)\(\sqrt[3]{(\mathrm{xy})^8}...
GMAT Advanced Math : (Adv_Math) Questions
Which expression is equivalent to \((\mathrm{xy})^{\frac{8}{3}}\), where x and y are positive?
- \(\sqrt[8]{(\mathrm{xy})^3}\)
- \(\sqrt[3]{(\mathrm{xy})^8}\)
- \(\sqrt[8]{(\mathrm{xy})^8}\)
- \(\sqrt[3]{(\mathrm{xy})^3}\)
1. TRANSLATE the fractional exponent using the conversion rule
- Given expression: \((\mathrm{xy})^{8/3}\)
- Need to convert to radical form using: \(\mathrm{x}^{m/n} = \sqrt[n]{\mathrm{x}^m}\)
- This tells us we need to identify the base, numerator (m), and denominator (n)
2. INFER the components for the conversion
- Base = \((\mathrm{xy})\)
- Numerator of exponent = 8 (this becomes the power inside the radical)
- Denominator of exponent = 3 (this becomes the index of the root)
3. TRANSLATE to build the radical expression
- Start with the nth root symbol: \(\sqrt[3]{}\)
- Place the base with its new power inside: \(\sqrt[3]{(\mathrm{xy})^8}\)
- Check: The denominator 3 became the index, the numerator 8 became the power
Answer: B. \(\sqrt[3]{(\mathrm{xy})^8}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students mix up which part of the fraction goes where in the radical conversion.
They might think \(\mathrm{x}^{8/3}\) converts to \(\sqrt[8]{(\mathrm{xy})^3}\), reversing the roles of numerator and denominator. They remember there's a conversion rule but apply it backwards, putting the numerator (8) as the index and denominator (3) as the power.
This may lead them to select Choice A (\(\sqrt[8]{(\mathrm{xy})^3}\))
Second Most Common Error:
Incomplete TRANSLATE reasoning: Students partially remember the rule but don't complete the conversion properly.
They might recognize that 8 and 3 both need to appear in the radical form, but incorrectly place both as either the index or the power, leading to something like \(\sqrt[8]{(\mathrm{xy})^8}\) where they use the larger number (8) for both components.
This may lead them to select Choice C (\(\sqrt[8]{(\mathrm{xy})^8}\))
The Bottom Line:
This problem tests precise knowledge of the fractional exponent conversion rule. Success requires not just knowing the rule exists, but remembering exactly which component of the fraction becomes which part of the radical.