What is the value of i raised to the 97th power (i⁹⁷)?

To find the value of i⁹⁷, let’s first clarify what the imaginary unit i is. The imaginary unit i is defined as the square root of -1. When we raise i to various powers, it follows a recurring pattern:

• i¹ = i
• i² = -1
• i³ = -i
• i⁴ = 1

This pattern repeats every four powers, which means for higher powers of i, we can use the modulus of the power with respect to 4 to simplify our calculations. In this case, we need to find 97 modulo 4:

97 ÷ 4 = 24 with a remainder of 1.

Since the remainder is 1, we can say:

i⁹⁷ = i^{(4 * 24) + 1} = (i⁴)²⁴ * i¹ = 1²⁴ * i = i

Thus, the value of i raised to the 97th power is:

Result: i

This shows that i⁹⁷ is simply equal to i.