Question

Which of the following can be the value of $\sqrt{4+3\sqrt{-20}}$ ?

• A. $(5+\sqrt{3}i)$
• B. $(3+\sqrt{5}i)$
• C. $(5-\sqrt{3}i)$
• D. $(3-\sqrt{5}i)$

Answer 1

The correct option is B $(3+\sqrt{5}i)$

Given: $\sqrt{4+3\sqrt{-20}}$
$\Rightarrow \sqrt{4+6\sqrt{5}i}$

Let, $\sqrt{4+6\sqrt{5}i}=x+iy...\left(i\right)$
squaring on both the sides of $\left(i\right)$ we get,
$4+6\sqrt{5}i=(x+iy{)}^{2}4+6\sqrt{5}i=x^2-y^2+2xyi...(ii)$
On comparing real and imaginary parts on both sides of $\left(ii\right)$, we get
$x^2-y^2=4\text{and}2xy=6\sqrt{5}$
$\Rightarrow x^2-y^2=4\text{and}xy=3\sqrt{5}$
$\Rightarrow x^2+y^2=\sqrt{(x^2-y^2{)}^2+4x^2{y}^2}=\sqrt{(4{)}^2+4(3{)}^{2}5}$
$\Rightarrow x^2+y^2=\sqrt{196}=14$

$\Rightarrow x^2-y^2=4\text{and}{x}^2+y^2=14$
$\Rightarrow x^2=9\text{and}{y}^2=5$
$\Rightarrow x=+3,-3\text{and}y=+\sqrt{5},-\sqrt{5}$

Since $xy>0,$ so $x\text{and}y$ are of the same sign
$\therefore (x=3,y=\sqrt{5})\text{or}(x=-3,y=-\sqrt{5})$

$\text{Hence,}\sqrt{4+3\sqrt{-20}}=(3+\sqrt{5}i)\text{or}(-3-\sqrt{5}i)$