Questions: Find the value of the expression. Simplify fully. i^5

Solution Steps

To find the value of \( i^5 \), we need to use the properties of the imaginary unit \( i \), where \( i^2 = -1 \). We can express higher powers of \( i \) in terms of \( i, -1, \) and \( -i \) by recognizing the cyclical pattern: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \), and then it repeats. Thus, \( i^5 = i^{4+1} = i^4 \cdot i^1 = 1 \cdot i = i \).

The imaginary unit \( i \) is defined such that \( i^2 = -1 \). The powers of \( i \) follow a cyclical pattern:

• \( i^1 = i \)
• \( i^2 = -1 \)
• \( i^3 = -i \)
• \( i^4 = 1 \)
• \( i^5 = i \) (since \( i^5 = i^{4+1} = i^4 \cdot i = 1 \cdot i \))

Using the cyclical pattern, we find that: \[ i^5 = i \]

Final Answer