Questions: Perform the indicated operation, and write the expression in the standard f i^-60 i^-60= (Simplify your answer.)

Transcript text: Perform the indicated operation, and write the expression in the standard $f$ \[ i^{-60} \] \[ i^{-60}=\square \] (Simplify your answer.)

Solution

Solution Steps

To simplify $ i^{-60} $, we need to understand the properties of the imaginary unit $ i $, where $ i^2 = -1 $. The powers of $ i $ cycle every four: $ i^1 = i $, $ i^2 = -1 $, $ i^3 = -i $, $ i^4 = 1 $, and then it repeats. To find $ i^{-60} $, we can use the property $ i^n = i^{n \mod 4} $ and the fact that $ i^{-n} = \frac{1}{i^n} $.

Step 1: Understanding the Powers of $ i $

The imaginary unit $ i $ has a cyclical pattern in its powers:

$ i^1 = i $
$ i^2 = -1 $
$ i^3 = -i $
$ i^4 = 1 $

This cycle repeats every four powers. Therefore, for any integer $ n $, $ i^n = i^{n \mod 4} $.

Step 2: Simplifying $ i^{-60} $

To simplify $ i^{-60} $, we first find the equivalent positive power by using the property $ i^{-n} = \frac{1}{i^n} $. We calculate $ n \mod 4 $ for $ n = -60 $.

\[ -60 \mod 4 = 0 \]

Thus, $ i^{-60} = \frac{1}{i^0} = \frac{1}{1} = 1 $.