Question

simplify.
i^{68}
type your answer in the blank below.

Explanation:

Step1: Recall the pattern of \(i\) powers

The imaginary unit \(i\) has a cyclic pattern for its powers: \(i^1 = i\), \(i^2=-1\), \(i^3 = i^2\times i=-i\), \(i^4=(i^2)^2 = (-1)^2 = 1\), and then the pattern repeats every 4 powers.

Step2: Divide the exponent by 4

To simplify \(i^{68}\), we divide the exponent 68 by 4. \(68\div4 = 17\) with a remainder of 0.

Step3: Use the pattern

Since the remainder is 0, this means \(i^{68}=(i^4)^{17}\). We know that \(i^4 = 1\), so \((i^4)^{17}=1^{17}=1\).

Answer:

1