what is the value of $i^{97}-i$?\n$-i$\n$0$\n$-2i$\n$i^{96}$
what is the value of $i^{97}-i$?\n$-i$\n$0$\n$-2i$\n$i^{96}$
Answer
Explanation:
Step1: Recall the powers of $i$
The powers of the imaginary unit $i$ have a cycle: $i^1 = i$, $i^2=- 1$, $i^3 = i^2\times i=-i$, $i^4=(i^2)^2 = 1$.
Step2: Divide the exponent by 4
Divide 97 by 4: $97\div4 = 24\cdots\cdots1$. So $i^{97}=(i^4)^{24}\times i^1$.
Step3: Simplify $i^{97}$
Since $i^4 = 1$, then $(i^4)^{24}=1^{24}=1$, and $i^{97}=1\times i = i$.
Step4: Calculate $i^{97}-i$
Substitute $i^{97}=i$ into the expression: $i - i=0$.
Answer:
0