Evaluating Powers of the Imaginary Unit '$i$'

The question asks for the value of the expression $\frac{1}{i^n}$, where $i = \sqrt{-1}$ and $n$ is specified as an even positive integer.

Understanding the Powers of '$i$'

The powers of the imaginary unit $i$ follow a cycle:

• $i^1 = i$
• $i^2 = -1$
• $i^3 = i^2 \cdot i = -1 \cdot i = -i$
• $i^4 = i^2 \cdot i^2 = (-1) \cdot (-1) = 1$
• $i^5 = i^4 \cdot i = 1 \cdot i = i$ (The cycle repeats every 4 powers)

Simplifying the Expression $\frac{1}{i^n}$ for Even '$n$'

Since $n$ is an even positive integer, we can write $n$ in the form $n = 2k$, where $k$ is a positive integer ($k = 1, 2, 3, \dots$).

Let's analyze the expression $i^n$ using this form:

$i^n = i^{2k} = (i^2)^k = (-1)^k$

Now, substitute this back into the original expression:

$\frac{1}{i^n} = \frac{1}{(-1)^k}$

Determining Possible Values

The value of $\frac{1}{(-1)^k}$ depends on whether $k$ is even or odd.

• Case 1: $k$ is odd. For example, if $k=1$, then $n=2$. In this case, $(-1)^k = -1$. The expression becomes $\frac{1}{-1} = -1$.
• Case 2: $k$ is even. For example, if $k=2$, then $n=4$. In this case, $(-1)^k = 1$. The expression becomes $\frac{1}{1} = 1$.

Since $n$ can be any even positive integer (like 2, 4, 6, 8, ...), $k$ can be any positive integer (like 1, 2, 3, 4, ...). Therefore, $k$ can be either odd or even.

Consequently, the value of $\frac{1}{i^n}$ can be either $+1$ or $-1$.

Final Result

The possible values for $\frac{1}{i^n}$ when $n$ is an even positive integer are $+1$ or $-1$. This corresponds to Option A.