Value of Iota to the Power of 6

The value of $i^6$i6 can be determined by understanding the powers of the imaginary unit $i$i, where $i$i is defined as the square root of $-1$−1. To find $i^6$i6, we can use the properties of $i$i in complex numbers. The imaginary unit $i$i has a cyclical pattern in its powers:

1. $i^1=i$i1=i
2. $i^2=-1$i2=−1
3. $i^3=-i$i3=−i
4. $i^4=1$i4=1
5. $i^5=i$i5=i
6. $i^6=-1$i6=−1

The cycle repeats every four powers. Thus, $i^6$i6 equals $-1$−1. This pattern arises because $i^4=1$i4=1 and any higher power can be reduced by expressing it as a multiple of 4 plus a remainder. Hence, $i^6$i6 can be simplified by the following steps:

1. Express 6 as $4+2$4+2.
2. Apply $i^4=1$i4=1 and $i^2=-1$i2=−1.
3. Therefore, $i^6=i^{4+2}=i^4\cdot i^2=1\cdot (-1)=-1$i6=i4+2=i4⋅i2=1⋅(−1)=−1.

This result is consistent with the cyclic nature of the imaginary unit's powers.

In summary, the value of $i^6$i6 is $-1$−1, and understanding the cyclical pattern of powers of $i$i simplifies the process of computing higher powers of the imaginary unit.