Question

1. evaluate: (i^{11})

a. (-1) b. (i) c. (1) d. (-i)

Explanation:

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

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

Step2: Divide the exponent by 4

We divide 11 by 4: \(11 \div 4 = 2\) with a remainder of 3. So, \(i^{11}=i^{4\times2 + 3}\).

Step3: Use the property of exponents

Using the property \(a^{m + n}=a^m \cdot a^n\), we can write \(i^{4\times2 + 3}=(i^4)^2 \cdot i^3\).
Since \(i^4 = 1\) and \(i^3=-i\), we substitute these values: \((1)^2 \cdot (-i)= -i\).

Answer:

D. \(-i\)