gmatophobia wrote:
\(-(\frac{1}{2})^{-\frac{1}{3}}, \quad -(\frac{1}{4})^{-\frac{1}{2}}, \quad -(\frac{1}{4})^{-\frac{2}{3}},\quad -(\frac{1}{3})^{-\frac{1}{2}}, \quad -(\frac{1}{4})^{-\frac{1}{3}}\)
What is the median of the list of numbers above?
A. \(-(\frac{1}{2})^{-\frac{1}{3}}\)
B. \(-(\frac{1}{4})^{-\frac{1}{2}}\)
C. \(-(\frac{1}{4})^{-\frac{2}{3}}\)
D. \(-(\frac{1}{3})^{-\frac{1}{2}}\)
E. \(-(\frac{1}{4})^{-\frac{1}{3}}\)
Attachment:
Screenshot 2024-01-01 181734.png
\(-(\frac{1}{2})^{-\frac{1}{3}}, \quad -(\frac{1}{4})^{-\frac{1}{2}}, \quad -(\frac{1}{4})^{-\frac{2}{3}},\quad -(\frac{1}{3})^{-\frac{1}{2}}, \quad -(\frac{1}{4})^{-\frac{1}{3}}\)
let us convert the negative power to positive power
\(-(2)^{\frac{1}{3}}, \quad -(4)^{\frac{1}{2}}, \quad -(4)^{\frac{2}{3}},\quad -(3)^{\frac{1}{2}}, \quad -(4)^{\frac{1}{3}}\)
Now let us raise all to same power 1/6
\(-(2^2)^{\frac{1}{6}}, \quad -(4^3)^{\frac{1}{6}}, \quad -(4^4)^{\frac{1}{6}},\quad -(3^3)^{\frac{1}{6}}, \quad -(4^2)^{\frac{1}{6}}\)
\(-(4)^{\frac{1}{6}}, \quad -(64)^{\frac{1}{6}}, \quad -(256)^{\frac{1}{6}},\quad -(27)^{\frac{1}{6}}, \quad -(16)^{\frac{1}{6}}\)
Now all raised to same power. So, if we discard '-' sign for this step, the increasing order will be
\((256)^{\frac{1}{6}}> \quad (64)^{\frac{1}{6}}> \quad (27)^{\frac{1}{6}}>\quad (16)^{\frac{1}{6}}> \quad (4)^{\frac{1}{6}}\)
All though the median will remain the same, let us change the sign by multiplying by - sign
\(-(256)^{\frac{1}{6}}< \quad -(64)^{\frac{1}{6}}< \quad -(27)^{\frac{1}{6}}<\quad -(16)^{\frac{1}{6}}< \quad -(4)^{\frac{1}{6}}\)
Answer: D. \(-(\frac{1}{3})^{-\frac{1}{2}}\)