Question

If \[\sqrt {x + iy} = \pm (a + ib)\] then what is the \[\sqrt { - x - iy} \] equal to ?

Answer 1

Hint: To find the solution of this problem, first of all you have to make a given equation similar to a given statement. Then you will see that iota is present in the next term so multiply it with the given statement and remember the logic of multiplying two iota and you will get the correct answer.

Complete answer:
Take (-1) as common in \[\sqrt {xy} = \sqrt x \times \sqrt y \] , so it becomes basically nothing but \[y \ne 0\] .
Now, as I mentioned in the hint, If \[\sqrt {xy} = \sqrt x \times \sqrt y \] , we can write it as here. So, \[ \pm i(a + ib)\]
Because the value of \[\sqrt { - 1} = i\]. And the value of \[\sqrt {(x + iy)} = \pm (a + ib)\] which is already given in this question.
So finally, It becomes \[ \pm (ai - b)\] by multiplying $i$ in this value and it gives the resultant output.
After that, we multiply $i$ with the \[(a + ib)\] , so it becomes \[ \pm (b - ia)\]. (Here the reason for changing the sign of b is that when two \[i\] multiply with each other, then it becomes (-1) = $i \times i$ . So that’s the reason.
Here there is \[ \pm \]sign, so we can take (-) sign as common and multiply with it, but it still remains \[ \pm \]. So, the final answer becomes \[ \pm (b - ia)\].

Note:
You can use this formula also \[\sqrt {x - iy} = \pm (\sqrt {(|z| + x)/2} - i\sqrt {(|z| - x)/2} )\] (when y > 0 and \[y \ne 0\] ).
 $\sqrt {x - iy} = \pm \sqrt {\dfrac{{(|z| + x)}}{2}} - i\sqrt {\dfrac{{(|z| - x)}}{2}} $ for ( y < 0 and \[y \ne 0\]). And also change the sign in the formula according to the sign of x in the question. So it basically depends on the value of x and y if we look carefully so you don’t need to remember any other formula of this type, this two is more than enough.