What happens to the value of i raised to odd powers?

When examining the powers of the imaginary unit ( i ), it is important to recognize the pattern that emerges as ( i ) is raised to successive powers. The fundamental property of ( i ) is that ( i^2 = -1 ). This relationship allows us to expand the powers of ( i ) systematically:

• ( i^1 = i )

• ( i^2 = -1 )

• ( i^3 = i^2 \cdot i = -1 \cdot i = -i )

• ( i^4 = (i^2)^2 = (-1)^2 = 1 )

• Continuing this pattern, the next powers repeat the sequence:

• ( i^5 = i )

• ( i^6 = -1 )

• ( i^7 = -i )

• ( i^8 = 1 )

From this information, we see that the powers of ( i ) form a cyclical pattern repeating every four powers: ( i, -1, -i, 1 ). Focusing specifically on the odd powers, we get:

• For ( i^1 ), the result is ( i \