Final answer:
The number \(\frac{1}{64}\) has six real sixth roots because it simplifies to \(2^{-6}\), which has one positive and one negative real sixth root, along with their complex conjugates, totaling six roots. The correct answer to the question is \(C) 6\)
Step-by-step explanation:
The question asks about the number of real sixth roots of the number \(\frac{1}{64}\). A real sixth root of a number is a number that, when raised to the power of six, would equal the original number. The sixth root of a number can be positive or negative, but it must be a real number.
\(\frac{1}{64}\) simplifies to \(2^{-6}\) since \(64\) is \(2^6\). This means we are looking for numbers that when raised to the power of six equal \(2^{-6}\). The most obvious root is \(2^{-1}\), which is \(\frac{1}{2}\), and its negative counterpart \(-2^{-1}\), which is \(-\frac{1}{2}\). However, because the original number is positive and the power is even, the roots will also include their complex conjugates, resulting in a total of six roots: three pairs of conjugate complex numbers.
Therefore, the correct answer to the question is \(C) 6\), indicating that there are six real sixth roots of the number \(\frac{1}{64}\)