**Fourth roots of unity power**, take variable **x** equal to 1 root’s power** 4** this gives the 4 roots **+1, -1, +ι, -ι**

**X = ∜ 1**

**X= (1)¼**

**Suppose that**

**x be a fourth root of unity**

Therefore

**x=∜1**

taking power 4 on both side

**x⁴=(1)⁴×¼**

after cancelation 4×1/4=1

one power 1 =1

we get

**x⁴=1**

**X⁴ −1=0**

we know the algebraic formula a²-b²=(a-b)(a+b)

**⇒ (x² – 1)(x² + 1) = 0**

⇒ (x² – 1) =0

x² = 1

**x = ±1…………………1**

⇒ (x² + 1) = 0

x² = – 1

**x = ± ι……………………..2**

Hence, the fourth root of unity are:

**+1, – 1, + ι, – ι**

**Properties of four fourth roots of unity power 4**

we have found that the four fourth root of unity are variable x power 4 = 1

**+1, – 1, + ι, – ι**

**The sum of all the four fourth roots of unity power is zero**

**+1+(-1)+ι + (-ι) = 0**

**+1-1+ι-ι =0**

**Real fourth roots of unity are additive inverse of each other **

**+ 1 and – 1 are the real fourth root of unity**

and

+ 1 + (-1) =0 = (-1) +1 (additive inverse of each other)

**Complex or imaginary fourth roots of unity are conjugate of each other:**

**iota and – iota** are complex or imaginary fourth root of unity

**0 – ι conjugate equal to 0 + ι**

Which is obviously conjugate of each other

**Product of all the fourth Roots of unity is – 1 **

**1 × (-1) × ι × (- ι) = -1**

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