Which of the following numbers is rational?
\(\sqrt[7]{128}\)
Write 128 as a power of 2: \(128 = 2^7\).
The \(n\)th root of 128 is \(2^{7/n}\); this is rational only when \(7/n\) is an integer.
Among the given roots, only \(n = 7\) gives an integer exponent, so \(128^{1/7} = 2\).
The others give irrational values \(2^{7/3}\), \(2^{7/5}\) and \(2^{7/4}\).
Hence, \(\sqrt[7]{128}\).
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