The powers of the imaginary number i have four possible values: 1, i, -1, and -i. Let's see some examples:
iā° = 1 (any number with the exponent 0 equals 1)
iĀ¹ = i (any number with the exponent 1 is the number itself)
iĀ² = -1 (this follows from the definition of the imaginary number, the square root of -1)
iĀ³ = iĀ² * i = -1 * i = -i
Now, the results start to repeat from 1 to -i:
iā“= iĀ² * iĀ² = (-1) * (-1) = 1
iāµ = iā“ * i = 1 * i = i
iā¶ = i * iāµ = i * i = iĀ² = -1
iā· = iā¶ * i = -1 * i = -i
From that, we can follow the steps below to find the value of a power of i:
ā¢ divide the exponent by 4;
,ā¢ if the rest of the division is 0, then the power equals ,iā° = 1,;
,ā¢ if the rest of the division is 1, then the power equals ,iĀ¹ = i,;
,ā¢ if the rest of the division is 2, then the power equals ,iĀ² = -1,;
,ā¢ if the rest of the division is 3, then the power equals ,iĀ³ = -i,.
So, for the power iā¶ā“ā¹, the exponent is 649. Following the steps above, we find:
ā¢ 649/4, has a quotient of 162 and the, rest 1,. Therefore:
iā¶ā“ā¹ = iĀ¹ = i
Thus, iā¶ā“ā¹ equals i.
