Find the value of $√(4+6\sqrt{5}i) + √(4-6\sqrt{5}i)$

Question

Find the value of $\sqrt{(4+6\sqrt{5}i)} + \sqrt{(4-6\sqrt{5}i)}$

$\sqrt{(4+6\sqrt{5}i)} = ± (3+\sqrt{5}i)$

$\sqrt{(4-6\sqrt{5}i)} = ± (3-\sqrt{5}i)$

There are two solutions to each which implies on adding them together, we will get four combinations and four different answers.

These answers are $6,-6,-2\sqrt{5}i,2\sqrt{5}i$.

But the only answer given in my book is $6$; they have only taken the + and + combination. Why we don't take the remaining three combinations? Why aren't all four the answers?

The two complex numbers you are adding are conjugates of each other, hence the sum must be real. — Anurag A, May 29 '17 at 17:45
Typically the $n $th root of a non-real complex number is considered to be that with the largest positive real part. — Oscar Lanzi, May 29 '17 at 17:53
Cotton: in general for such problems the concept of principle square root of a complex number is intended. For more details, check https://math.stackexchange.com/questions/44406/how-do-i-get-the-square-root-of-a-complex-number — Anurag A, May 29 '17 at 17:55

score 0 · Answer 1 · answered May 29 '17 at 17:53

0

I think the book means the following. $$\left(\sqrt{4+6\sqrt5i}+\sqrt{4-6\sqrt5i}\right)^2=8+2\cdot14=36$$ and the book gets $\sqrt{4+6\sqrt5i}+\sqrt{4-6\sqrt5i}=6$.

This reasoning is wrong, of course.

answered May 29 '17 at 17:53

Michael Rozenberg

194,933

No it's just taxonomy, basically, see the comments. – Oscar Lanzi May 29 '17 at 17:55
@Oscar Lanzi I think the answer of Cotton is true. I tried to show a wrong solution from the book. – Michael Rozenberg May 29 '17 at 18:03

Find the value of $√(4+6\sqrt{5}i) + √(4-6\sqrt{5}i)$

1 Answers1