Solved on Mar 14, 2024

Simplify the complex number i28i^{-28} to its most basic form.

STEP 1

Assumptions
1. The complex number ii is defined such that i2=1i^2 = -1.
2. The exponent 28-28 is a negative even integer.

STEP 2

We know that i2=1i^2 = -1. We can use this to find patterns for higher powers of ii.
i2=1i^2 = -1 i3=i2i=ii^3 = i^2 \cdot i = -i i4=i2i2=1i^4 = i^2 \cdot i^2 = 1

STEP 3

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

STEP 4

We will divide the exponent 28-28 by 4 and find the remainder.
28mod4-28 \mod 4

STEP 5

Calculate the remainder.
28mod4=0-28 \mod 4 = 0

STEP 6

Since the remainder is 0, i28i^{-28} is equivalent to i0i^0 because 28-28 is a multiple of 4.

STEP 7

Any non-zero number to the power of 0 is 1.
i0=1i^0 = 1

STEP 8

Therefore, the simplified form of i28i^{-28} is 1.
i28=1i^{-28} = 1

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