Question

which of the following is equivalent to the complex number $i^{11}$? choose 1 answer: a 1 b $i$ c $-1$ d $-i$

Explanation:

Step1: Recall the cycle of \(i\)

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 cycle repeats every 4 powers. So we can find the remainder when the exponent is divided by 4 to simplify \(i^{11}\).

Step2: Divide 11 by 4

Divide 11 by 4: \(11 \div 4 = 2\) with a remainder of 3. This means \(i^{11}=i^{4\times2 + 3}\).

Step3: Use the property of exponents

Using the property \(a^{m + n}=a^m \cdot a^n\), we can rewrite \(i^{4\times2 + 3}\) as \((i^4)^2 \cdot i^3\).

Step4: Substitute the known values

We know \(i^4 = 1\) and \(i^3=-i\). So \((i^4)^2 \cdot i^3=(1)^2 \cdot (-i)=-i\).

Answer:

D. \(-i\)