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then ${z_1}^2 + {z_2}^2 + {z_3}^2 + {z_4}^2$ is equal to

A.$1$

B.$0$

C.$i$

D.None of these

Answer
Verified

Fourth roots of Unity

Properties of Four Fourth Roots of Unity

a. Sum of all the four fourth roots of unity is zero.

b. The real fourth roots of unity are additive

Inverse of each other.

c. Both the complex / imaginary Fourth roots of

unity are conjugate for each other

d. Product of all the Fourth roots of unity is â€“

Let $x$ be the four fourth roots of $1$, if then we can write

$x = 4\sqrt 1 $

We should write it

$x = {(1)^{\dfrac{1}{4}}}$

$ \Rightarrow {x^4} = 1$

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

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

\[[{a^2} - {b^2} = (a + b)(a - b)]\]

Therefore,

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

Either,

\[({x^2} - 1) = 0 or ({x^2} + 1) = 0\]

\[{x^2} = 1 or {x^2} = - 1\]

\[x = \pm \sqrt 1 or x = \pm \sqrt { - 1} \]

\[x = \pm 1 or x = \pm i\]

Now, the Four fourth roots are unity is $[1, - 1,i, - i]$

Now we complete the answer

Step by step

(Image)

\[{z_1},{z_2},{z_3},{z_4}\] are roots of

${x^4} - 1 = 0$

\[\therefore {z_1} + {z_2} + {z_3} + {z_4} = 0\]

\[{z_1}{z_2} + {z_2}{z_3} + {z_3}{z_4} + {z_4}{z_1} + {z_1}{z_3} + {z_2}{z_4} = 0\]

\[\therefore {({z_1} + {z_2} + {z_3} + {z_4})^2} = \sum\limits_{}^{} {{z_1}^2} \]

$\sum\limits_{i = 1}^4 {} \sum\limits_{i = 1}^4 {} {z_1}{z_i}$

\[0 = {\sum {{z_1}} ^2} = 0\]

\[\therefore {\sum {{z_1}} ^2} = 0\]

B=$0$

Therefore, the combination of both real and imaginary numbers is a complex number.