x+ix4+x2+1=122x+i 2x2+x+1 x2-x+1 =12(x2+x+1)-(x2-x+1)+i2x2+x+1 x2-x+1 =12x2+x+1+i x2-x+12 =x2+x+1+ix2-x+12

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$\sqrt{x + i \sqrt{x^{4} + x^{2} + 1}} =$

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$\frac{\sqrt{x^{2} + x + 1} + i \sqrt{x^{2} - x + 1}}{\sqrt{2}}$

$\frac{\sqrt{x^{2} + x + 1} - i \sqrt{x^{2} - x + 1}}{2}$

$\sqrt{x + 1} + i \sqrt{x - 1}$

$\sqrt{x - 1} + i \sqrt{x + 1}$

answer is A.

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Detailed Solution

$\sqrt{x + i \sqrt{x^{4} + x^{2} + 1}} = \frac{1}{\sqrt{2}} \sqrt{2 x + i 2 \sqrt{x^{2} + x + 1} \sqrt{x^{2} - x + 1}} = \frac{1}{\sqrt{2}} \sqrt{(x^{2} + x + 1) - (x^{2} - x + 1) + i 2 \sqrt{x^{2} + x + 1} \sqrt{x^{2} - x + 1}} = \frac{1}{\sqrt{2}} \sqrt{{(\sqrt{x^{2} + x + 1} + i \sqrt{x^{2} - x + 1})}^{2}} = \frac{\sqrt{x^{2} + x + 1} + i \sqrt{x^{2} - x + 1}}{\sqrt{2}}$