Question

er of i.

1. ( i^7 )

Explanation:

Step1: Recall the powers of \(i\)

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

Step2: Divide the exponent by 4

For \(i^7\), divide 7 by 4. \(7 \div 4 = 1\) with a remainder of 3. So, \(i^7 = i^{4 \cdot 1 + 3} = (i^4)^1 \cdot i^3\).

Step3: Substitute the known powers

We know \(i^4 = 1\) and \(i^3 = -i\). So, \((i^4)^1 \cdot i^3 = 1 \cdot (-i) = -i\).

Answer:

\(-i\)