What is the value of i raised to the power of 63?

To determine the value of (i) raised to the power of 63, it is important to understand the cyclical nature of the powers of the imaginary unit (i). The powers of (i) cycle through a pattern every four exponents:

• (i^1 = i)

• (i^2 = -1)

• (i^3 = -i)

• (i^4 = 1)

Once you reach (i^4), the cycle repeats, so (i^5) would be the same as (i^1), (i^6) as (i^2), and so forth.

To find (i^{63}), you can reduce the exponent 63 modulo 4. This is because the powers of (i) repeat every four integers:

[

63 \mod 4 = 3

]

This calculation reveals that (i^{63}) equals the same as (i^3). From the established cycle, we know:

[

i^3 = -i

]

Therefore, the value of (i^{63}) is (-i), making that the correct answer.