Question

find the fourth root of unity

Answer 1

Here is your answer mate..✌✌✔

A complex number z such that z 4=1. There are 4 fourth roots of unity and they are 1, i,−1 and−i.

i is the fourth root of unity as it follows 4n exponents

Answer 2

Fourth root of unity ($\sqrt[4]{1}$) are $1$, $(-1)$, $i$ and $(-i)$, where $i=\sqrt{-1}$.

Step-by-step explanation:

Let us take: $x^{4}=1$

$\Rightarrow x^{4}-1=0$

• transposing $1$ to the left hand side

$\Rightarrow (x^{2})^{2}-(1)^{2}=0$

• exponent rule: $a^{2m}=(a^{m})^{2}$

$\Rightarrow (x^{2}+1)(x^{2}-1)=0$

• using algebraic identity: $a^{2}-b^{2}=(a+b)(a-b)$

$\Rightarrow (x^{2}-\overline{-1})(x^{2}-1)=0$

$\Rightarrow (x^{2}-i^{2})(x^{2}-1^{2})=0$

• since $i=\sqrt{-1}\Rightarrow i^{2}=-1$

$\Rightarrow (x+i)(x-i)(x+1)(x-1)=0$

• using algebraic identity: $a^{2}-b^{2}=(a+b)(a-b)$

So, either $x+i=0$ or $x-i=0$ or $x+1=0$ or $x-1=0$

$\Rightarrow x=-i,i,-1,1$

$\Rightarrow \boxed{x=1,-1,i,-i}$