What is the result of i raised to the even powers?

When considering the results of ( i ) raised to even powers, it is important to recognize how the imaginary unit ( i ) behaves with various exponentials. The complex number ( i ) is defined as the square root of -1, and its powers cycle through a specific pattern.

When ( i ) is raised to the first power, it equals ( i ). When raised to the second power, it equals ( i^2 ), which simplifies to -1. Continuing this pattern:

• ( i^3 ) equals ( i^2 \cdot i ), which is -1 times ( i ) or -i.

• ( i^4 ) equals ( (i^2)^2 ), which is ((-1)^2 = 1).

From this sequence, we can observe that:

• ( i^0 = 1 ) (any number to the power of zero equals one)

• ( i^2 = -1 )

• ( i^4 = 1 )

• ( i^6 = -1 )

• ( i^8 = 1 )

It becomes evident that if ( n ) is an even number, ( i^