Question

simplify the power of $i$.
$i^{101}=$

Explanation:

Step1: Recall the cycle of powers of $i$

$i^1 = i$, $i^2=- 1$, $i^3 = i^2\times i=-i$, $i^4=(i^2)^2 = 1$. The powers of $i$ repeat every 4.

Step2: Divide the exponent by 4

Divide 101 by 4: $101\div4 = 25$ with a remainder. Using the division - formula $a = bq + r$, where $a = 101$, $b = 4$, $q = 25$, and $r=1$. So $101 = 4\times25+1$.

Step3: Rewrite $i^{101}$

$i^{101}=i^{4\times25 + 1}$. According to the exponent - rule $a^{m + n}=a^m\times a^n$ and $(a^m)^n=a^{mn}$, we have $i^{4\times25+1}=(i^4)^{25}\times i^1$.

Step4: Simplify the expression

Since $i^4 = 1$, then $(i^4)^{25}=1^{25}=1$. So $(i^4)^{25}\times i^1=1\times i=i$.

Answer:

$i$