Question

Find the square root of $5+12\sqrt{-1}$

• A. $\pm (3+2\sqrt{-1})$
• B. $\pm (2-3\sqrt{-1})$
• C. $\pm (4+3\sqrt{-1})$
• D. None of these

Answer 1

The correct option is A $\pm (3+2\sqrt{-1})$

Given $z=5+12\sqrt{-1}=5+12i$.

Suppose that $(a+bi)$ is a square root of $(5+12i)$.

Then, $(a+bi{)}^2=(a^2-b^2)+(2ab)i=5+12i.$

Equate real and imaginary parts:

$a^2-b^2=5$

$2ab=12\Rightarrow b=\frac{6}{a}.$

So, $a^2-{\left(\frac{6}{a}\right)}^2=5$

$=>a^2-\frac{36}{a^2}=5$

$=>a^4-5a^2-36=0.$

$=>(a^2-9)(a^2+4)=0.$

Since a must be real, $a=3$ or $-3$.

This gives $b=2$ or $-2$, respectively.

Thus, we have two square roots: $3+2i$ or $-3-2i$

So $(a+bi)=\pm (3+2\sqrt{-1})$