Question

Evaluate: $\sqrt{6+8i}$

Answer 1

$\text{Let}\sqrt{6+8i}=(x+iy).\text{On squaring both sides of}\left(i\right),weget6+8i=(x+iy{)}^2\Rightarrow 6+8i=(x^2-y^2)+i(2xy).\text{On comparing real parts and imaginary parts on both sides of (ii) we get}{x}^2-y^2=6\text{and}2xy=8\Rightarrow x^2-y^2=6\text{and}xy=4\Rightarrow (x^2+y^2)=\sqrt{(x^2-y^2{)}^2+4x^2{y}^2}=\sqrt{6^2+4\times 16}=\sqrt{100}=10\Rightarrow x^2-y^2=6and{x}^2+y^2=10\Rightarrow 2x^2=16and2y^2=4\Rightarrow x^2=8and{y}^2=2\Rightarrow =\pm 2\sqrt{2}andy=\pm \sqrt{2}.\text{x =}\pm 2\sqrt{2}andy=\pm \sqrt{2}\text{.}$
$\text{since xy > 0, so x and y are of the same sign.}\therefore (x=2\sqrt{2}\text{and}y=\sqrt{2})\text{or}(x=-2\sqrt{2}andy=-\sqrt{2}.\text{Hence},\sqrt{6+8i}=(2\sqrt{2}+\sqrt{2}i\left)or\right(-2\sqrt{2}-\sqrt{2}i.)$