Question

compute $i^{350}$.

Explanation:

Step1: Recall the cycle of \(i\)

The imaginary unit \(i\) has a cyclic pattern: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then the cycle repeats every 4 powers. So we can find the remainder when the exponent is divided by 4 to determine the equivalent power of \(i\).

Step2: Divide 350 by 4

We calculate \(350\div4\). \(4\times87 = 348\), so the remainder is \(350 - 348 = 2\). This means \(i^{350}\) is equivalent to \(i^{4\times87 + 2}\), and by the properties of exponents, \(i^{4\times87 + 2}=(i^4)^{87}\times i^2\).

Step3: Substitute the known values of \(i^4\) and \(i^2\)

We know that \(i^4 = 1\) and \(i^2 = -1\). So \((i^4)^{87}\times i^2 = 1^{87}\times(-1)\). Since \(1\) to any power is \(1\), this simplifies to \(1\times(-1)= -1\).

Answer:

\(-1\)