Question: 1 — Imaginary and Complex Numbers
LEARNING OBJECTIVE: Determine the value of a power of $i$.
Which of the following is equivalent to $i^{14}$?
a.) 1
b.) -1
c.) i
d.) -i
1. **State the problem:** We need to find the value of $i^{14}$, where $i$ is the imaginary unit with the property $i^2 = -1$.
2. **Recall the powers of $i$ cycle:** The powers of $i$ repeat every 4 steps:
$$i^1 = i$$
$$i^2 = -1$$
$$i^3 = -i$$
$$i^4 = 1$$
Then the cycle repeats: $i^5 = i$, $i^6 = -1$, and so on.
3. **Find the remainder when 14 is divided by 4:**
$$14 \div 4 = 3 \text{ remainder } 2$$
So,
$$14 = 4 \times 3 + 2$$
4. **Use the remainder to simplify:**
$$i^{14} = i^{4 \times 3 + 2} = (i^4)^3 \times i^2$$
Since $i^4 = 1$, this becomes:
$$= 1^3 \times i^2 = i^2$$
5. **Recall $i^2 = -1$:**
Therefore,
$$i^{14} = -1$$
**Final answer:** $i^{14}$ is equivalent to $-1$, which corresponds to option b.) -1.