Question

compute (i^{669}).

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 669 by 4: $669\div4 = 167$ with a remainder. Using the division formula $a = bq + r$, where $a = 669$, $b = 4$, $q = 167$ and $r=1$. So $669=4\times167 + 1$.

Step3: Rewrite $i^{669}$

We know that $i^{669}=i^{4\times167 + 1}$. According to the exponent - rule $a^{m + n}=a^m\times a^n$ and $(a^m)^n=a^{mn}$, we have $i^{4\times167+1}=(i^4)^{167}\times i^1$.

Step4: Simplify the expression

Since $i^4 = 1$, then $(i^4)^{167}=1^{167}=1$. So $(i^4)^{167}\times i^1=1\times i=i$.

Answer:

$i$