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Solved on Mar 14, 2024

Simplify the complex number $i^{-28}$ to its most basic form.

STEP 1

Assumptions
1. The complex number $i$ is defined such that $i^2 = -1$ .
2. The exponent $-28$ is a negative even integer.

STEP 2

We know that $i^2 = -1$ . We can use this to find patterns for higher powers of $i$ .
$i^2 = -1$ $i^3 = i^2 \cdot i = -i$ $i^4 = i^2 \cdot i^2 = 1$

STEP 3

Notice that the powers of $i$ repeat every 4 steps. This is a cycle: $i, -1, -i, 1$ . So, any power of $i$ can be reduced by taking the remainder when the exponent is divided by 4.

STEP 4

We will divide the exponent $-28$ by 4 and find the remainder.
$-28 \mod 4$

STEP 5

Calculate the remainder.
$-28 \mod 4 = 0$

STEP 6

Since the remainder is 0, $i^{-28}$ is equivalent to $i^0$ because $-28$ is a multiple of 4.

STEP 7

Any non-zero number to the power of 0 is 1.
$i^0 = 1$

STEP 8

Therefore, the simplified form of $i^{-28}$ is 1.
$i^{-28} = 1$

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