Question

which answer is equal to $i^{45}$?

• $i$
• $i$
• 1
• 1

Explanation:

Step1: Recall the powers of $i$ cycle

The powers of the imaginary unit $i$ have a cycle: $i^1 = i$, $i^2=- 1$, $i^3 = i^2\times i=-i$, $i^4=(i^2)^2 = 1$. The cycle repeats every 4 powers.

Step2: Divide the exponent by 4

Divide 45 by 4: $45\div4 = 11$ with a remainder. Using the division - formula $a = bq + r$, where $a = 45$, $b = 4$, $q = 11$ and $r=1$. So $i^{45}=i^{4\times11 + 1}$.

Step3: Apply the exponent - rule

According to the rule $(a^m)^n=a^{mn}$ and $a^{m + n}=a^m\times a^n$, we have $i^{4\times11+1}=(i^4)^{11}\times i^1$.

Step4: Substitute the values

Since $i^4 = 1$, then $(i^4)^{11}=1^{11}=1$. So $(i^4)^{11}\times i^1=1\times i=i$.

Answer:

B. $i$