How to simplify i to a power?

Understand the Problem

The question is asking how to simplify an expression involving the imaginary unit i raised to a power. Understanding the powers of i (i.e., i^1, i^2, i^3, etc.) and their periodic nature is essential for simplification.

Answer

To simplify $i^n$, calculate $n \mod 4$ and use the corresponding case: $1$ for $i^0$, $i^1$ for $i$, $-1$ for $i^2$, and $-i$ for $i^3$.

Steps to Solve

Identify the powers of i The imaginary unit $i$ has a cyclical pattern in its powers:

$i^1 = i$
$i^2 = -1$
$i^3 = -i$
$i^4 = 1$

Notice that this cycle repeats every 4 powers.

Determine the exponent mod 4 To simplify $i^n$, we can find the equivalent power by taking the exponent modulo 4. For example, if we have $i^{10}$, we calculate: $$ 10 \mod 4 = 2 $$ Thus, $i^{10} = i^2$.
Substitute and simplify After finding the equivalent power of $i$, substitute that back into the expression. For example: If $i^{10} = i^2$, then: $$ i^{10} = -1 $$ since $i^2 = -1$.

The simplified value of $i^n$ depends on the exponent $n$. Specifically, it can be computed using:

If $n \mod 4 = 0$, then $i^n = 1$.
If $n \mod 4 = 1$, then $i^n = i$.
If $n \mod 4 = 2$, then $i^n = -1$.
If $n \mod 4 = 3$, then $i^n = -i$.

More Information

The powers of the imaginary unit $i$ cycle every four terms. In practice, this can significantly speed up calculations involving powers of $i$, especially in complex numbers.

Tips

Forgetting the cyclical pattern: Always remember that the powers of $i$ repeat every four terms. Not recognizing this can lead to incorrect simplifications.
Incorrectly calculating modulo: Always double-check the modulus calculation; it's easy to make an error that can throw off the whole problem.

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