Questions: Simplify the following expression. i^173

Solution Steps

To simplify \(i^{173}\), we need to use the properties of the imaginary unit \(i\), where \(i\) is defined as the square root of -1. The powers of \(i\) cycle every four: \(i, i^2, i^3, i^4\). Specifically, \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). We can use this cyclical pattern to simplify \(i^{173}\) by finding the remainder when 173 is divided by 4.

The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\). The powers of \(i\) follow a cyclic pattern: \[ \begin{align_} i^1 &= i, \\ i^2 &= -1, \\ i^3 &= -i, \\ i^4 &= 1. \end{align_} \] This cycle repeats every four powers.

To simplify \(i^{173}\), we need to find the remainder when 173 is divided by 4, because the powers of \(i\) repeat every 4 steps.

\[ 173 \div 4 = 43 \text{ remainder } 1 \]

So, \(173 \equiv 1 \pmod{4}\).

Since \(173 \equiv 1 \pmod{4}\), we have: \[ i^{173} = i^1 = i \]

Final Answer