Bunuel wrote:
\(-(\frac{1}{3})^{-\frac{1}{5}}\), \(-(\frac{1}{2})^{-\frac{1}{4}}\), \(-(\frac{1}{5})^{-\frac{2}{3}}\), \(-(\frac{1}{2})^{-\frac{1}{3}}\), \(-(-(\frac{1}{25})^{-\frac{1}{3}})\).
What is the range of the list of numbers above?
A. \(0\)
B. \(\sqrt[3]{25}\)
C. \(5\)
D. \(2\sqrt[3]{25}\)
E. \(10\)
\(-(\frac{1}{3})^{-\frac{1}{5}}\), \(-(\frac{1}{2})^{-\frac{1}{4}}\), \(-(\frac{1}{5})^{-\frac{2}{3}}\), \(-(\frac{1}{2})^{-\frac{1}{3}}\), \(-(-(\frac{1}{25})^{-\frac{1}{3}})\).
- \(-(\frac{1}{3})^{-\frac{1}{5}}\) ⇒ \(-(3)^{\frac{1}{5}}\)
- \(-(\frac{1}{2})^{-\frac{1}{4}}\) ⇒ \(-(2)^{\frac{1}{4}}\)
- \(-(\frac{1}{5})^{-\frac{2}{3}}\) ⇒ \(-(5)^{\frac{2}{3}}\)
- \(-(\frac{1}{2})^{-\frac{1}{3}}\) ⇒ \(-(2)^{\frac{1}{3}}\)
- \(-(-(\frac{1}{25})^{-\frac{1}{3}})\) ⇒ \(-(-(25)^{\frac{1}{3}})\) ⇒ \(-(-(5)^{\frac{2}{3}})\)
Among the five above numbers, \(-(-(5)^{\frac{2}{3}})\) is positive and can be represented as \((5)^{\frac{2}{3}}\), all other numbers are negative.
Among the negative numbers, the value of \(-(5)^{\frac{2}{3}}\) is the lowest. We can approximate the value of \(-(5)^{\frac{2}{3}}\)
\(-(5)^{\frac{2}{3}} = -(25)^{\frac{1}{3}} \approx -3\)
The values of the remaining three terms will be greater than -3, and the values will lie to the right of -3 on a number line.
- Lowest value ⇒ \(-(5)^{\frac{2}{3}}\)
- Highest value ⇒ \((5)^{\frac{2}{3}}\)
Range = Highest Value - Lowest Value
= \((5)^{\frac{2}{3}} -(-(5)^{\frac{2}{3}}) = 2 * (5)^{\frac{2}{3}} = 2 * (25)^{\frac{1}{3}}\)
Option D