Question

question 2
which answer is equal to $i^{45}$ ?
\\(\boldsymbol{\circ} -1\\)
\\(\boldsymbol{\circ} -i\\)
\\(\boldsymbol{\circ} 1\\)
\\(\boldsymbol{\circ} i\\)

Explanation:

Step1: Recall the property of imaginary unit \(i\)

The imaginary unit \(i\) has a cyclic property: \(i^1 = i\), \(i^2=- 1\), \(i^3=i^2\times i=-i\), \(i^4=(i^2)^2 = (-1)^2 = 1\), and then the cycle repeats every 4 powers. So we can use the formula \(i^{n}=i^{n\bmod{4}}\), where \(\bmod\) is the modulo operation.

Step2: Calculate the exponent modulo 4

We need to find \(45\bmod{4}\). Divide 45 by 4: \(45 = 4\times11+1\), so \(45\bmod{4}=1\).

Step3: Simplify \(i^{45}\)

Since \(i^{45}=i^{4\times11 + 1}=(i^{4})^{11}\times i^{1}\), and \(i^{4} = 1\), then \((i^{4})^{11}=1^{11}=1\), so \(i^{45}=1\times i=i\).

Answer:

\(i\) (the option with \(i\))