"Let $\alpha$ and $\beta$ be the roots of the equation x2 + (2i $-$ 1) = 0. Then, the value of |$\alpha$8 + $\beta$8| is equal to :"

Question

Accepted Answer

"

Given equation,

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

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

Let $\alpha$ and $\beta$ are the two roots of the equation.

As, we know roots of a equation satisfy the equation so

${\alpha ^2} = 1 - 2i$

and ${\beta ^2} = 1 - 2i$

$\therefore$ ${\alpha ^2} = {\beta ^2} = 1 - 2i$

$\therefore$ $|{\alpha ^2}| = \sqrt {{1^2} + {{( - 2)}^2}} = \sqrt {15}$

Now, $|{\alpha ^8} + {\beta ^8}|$

$|{\alpha ^8} + {\alpha ^8}|$

$= 2|{\alpha ^8}|$

$= 2|{\alpha ^2}{|^4}$

$= 2{\left( {\sqrt 5 } \right)^4}$

$= 2 \times 25$

$= 50$

"

Let $\alpha$ and $\beta$ be the roots of the equation x² + (2i $-$ 1) = 0. Then, the value of |$\alpha$⁸ + $\beta$⁸| is equal to :