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\(-(\frac{1}{3})^{-\frac{1}{5}}\), \(-(\frac{1}{2})^{-\frac{1}{4}}\), \(-(\frac{1}{5})^{-\frac{2}{3}}\), \(-(\frac{1}{2})^{-\frac{1}{3}}\), \(-(-(\frac{1}{25})^{-\frac{1}{3}})\).
What is the range of the list of numbers above?
A. \(0\)
B. \(\sqrt[3]{25}\)
C. \(5\)
D. \(2\sqrt[3]{25}\)
E. \(10\)
What is the range of the list of numbers above?
A. \(0\)
B. \(\sqrt[3]{25}\)
C. \(5\)
D. \(2\sqrt[3]{25}\)
E. \(10\)
06 Mar 2024, 13:42
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Official Solution:
\(-(\frac{1}{3})^{-\frac{1}{5}}\), \(-(\frac{1}{2})^{-\frac{1}{4}}\), \(-(\frac{1}{5})^{-\frac{2}{3}}\), \(-(\frac{1}{2})^{-\frac{1}{3}}\), \(-(-(\frac{1}{25})^{-\frac{1}{3}})\).
What is the range of the list of numbers above?
A. \(0\)
B. \(\sqrt[3]{25}\)
C. \(5\)
D. \(2\sqrt[3]{25}\)
E. \(10\)
The range of a list is the difference between the largest and smallest elements of the list. Hence, to calculate the range, we must first find the smallest and largest numbers of the list given. For that, let's start by rewriting the numbers in easier format:
Unlike other numbers, the fifth number, \(\sqrt[3]{25}\) is positive and thus obviously the largest.
Each of \(-\sqrt[5]{3}\), \(-\sqrt[4]{2}\), and \(-\sqrt[3]{2}\) are -1.something, while \(-\sqrt[3]{25}\) is a little bit more than -3. Thus, \(-\sqrt[3]{25}\) is the smallest one.
Therefore, the range is \( \sqrt[3]{25} - (-\sqrt[3]{25}) = 2\sqrt[3]{25}\)
Answer: D
\(-(\frac{1}{3})^{-\frac{1}{5}}\), \(-(\frac{1}{2})^{-\frac{1}{4}}\), \(-(\frac{1}{5})^{-\frac{2}{3}}\), \(-(\frac{1}{2})^{-\frac{1}{3}}\), \(-(-(\frac{1}{25})^{-\frac{1}{3}})\).
What is the range of the list of numbers above?
A. \(0\)
B. \(\sqrt[3]{25}\)
C. \(5\)
D. \(2\sqrt[3]{25}\)
E. \(10\)
The range of a list is the difference between the largest and smallest elements of the list. Hence, to calculate the range, we must first find the smallest and largest numbers of the list given. For that, let's start by rewriting the numbers in easier format:
\(-(\frac{1}{3})^{-\frac{1}{5}} = -(3)^{\frac{1}{5}} = -\sqrt[5]{3}\)
\(-(\frac{1}{2})^{-\frac{1}{4}} = -(2)^{\frac{1}{4}} = -\sqrt[4]{2}\)
\(-(\frac{1}{5})^{-\frac{2}{3}} = -(5)^{\frac{2}{3}} = -\sqrt[3]{5^2} = -\sqrt[3]{25} \)
\(-(\frac{1}{2})^{-\frac{1}{3}} = -(2)^{\frac{1}{3}} = -\sqrt[3]{2}\)
\(-(-(\frac{1}{25})^{-\frac{1}{3}}) = -(-(25)^{-\frac{1}{3}}) = -(-(\sqrt[3]{25}) = \sqrt[3]{25}\)
\(-(\frac{1}{2})^{-\frac{1}{4}} = -(2)^{\frac{1}{4}} = -\sqrt[4]{2}\)
\(-(\frac{1}{5})^{-\frac{2}{3}} = -(5)^{\frac{2}{3}} = -\sqrt[3]{5^2} = -\sqrt[3]{25} \)
\(-(\frac{1}{2})^{-\frac{1}{3}} = -(2)^{\frac{1}{3}} = -\sqrt[3]{2}\)
\(-(-(\frac{1}{25})^{-\frac{1}{3}}) = -(-(25)^{-\frac{1}{3}}) = -(-(\sqrt[3]{25}) = \sqrt[3]{25}\)
Unlike other numbers, the fifth number, \(\sqrt[3]{25}\) is positive and thus obviously the largest.
Each of \(-\sqrt[5]{3}\), \(-\sqrt[4]{2}\), and \(-\sqrt[3]{2}\) are -1.something, while \(-\sqrt[3]{25}\) is a little bit more than -3. Thus, \(-\sqrt[3]{25}\) is the smallest one.
Therefore, the range is \( \sqrt[3]{25} - (-\sqrt[3]{25}) = 2\sqrt[3]{25}\)
Answer: D
