Which of the following is equivalent to lol \( ^{36} \) ? a.) \( i \) b.) \( -i \) c.) 1 d.) -1

To find the equivalent of \( i^{36} \), we can use the fact that the powers of \( i \) (the imaginary unit) cycle every 4: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). Since \( 36 \mod 4 = 0 \), \( i^{36} \) is equivalent to \( i^0 = 1 \). Thus, the answer is c.) 1. Interestingly, this cycling of powers is a fun result in mathematics that not only applies to the imaginary unit but also to other roots of unity. In fact, powers of complex numbers exhibit similar cyclical behaviors, which can lead to fascinating discoveries in number theory and algebra.