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Chapter 2: Problem 20

If \(c>0\) and \(m\) and \(n\) are positive integers, which of the following is equivalent to \(c^{\frac{m}{n}}\) ? A) \(\frac{c^m}{c^n}\) B) \(c m-n\) C) \((\sqrt[m]{c})^n\) D) \((\sqrt[n]{c})^m\)

Short Answer

Expert verified
The short answer is: D) \((\sqrt[n]{c})^m\)

Step by step solution

01

Preliminary Observation

\(c^{\frac{m}{n}}\) can be rewritten as \(\sqrt[n]{c^m}\).
02

Comparing to Choices

Now let's compare this to the given choices: A) \(\frac{c^m}{c^n}\): This simplification is incorrect, as it divides the exponents rather than taking an nth root. B) \(c m-n\): This simplification is also incorrect, as it subtracts the two exponents rather than taking an nth root. C) \((\sqrt[m]{c})^n\): This simplification is very close to our observation, but the exponents m and n have been swapped. D) \((\sqrt[n]{c})^m\): This exactly matches our observation that \(c^{\frac{m}{n}} = \sqrt[n]{c^m}\). This is the correct answer. So the equivalent expression for \(c^{\frac{m}{n}}\) is: D) \((\sqrt[n]{c})^m\)

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