Consider I am having a complex number:
$x = 4i$.
- Squaring on both sides: $x^2 = -16$.
- Multiply by $-1$ on both sides: $-x^2 = 16$.
- Taking square root on both sides: $xi = 4$.
What I am doing wrong here?
Why $i$ is not in denominator?
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The last step should be $x_i= \pm 4$. Because the square root of a number is either its positive or negative value. Recognize that
$4 \times 4 = 16$
$-4 \times -4 = 16$.
However, now you can check your answer with the first line $x=4i$. $x \times i = 4i \times i = -4$. Thus $xi=-4$.
So far so good.
Here you need to exercise some care. $(-1)(x^2) = -(x^2)$. Then the next step needs to be amended:
Okay, wait, where exactly are you going with this? You need to be clear on the difference between the principal square root and the other square root. If you start with $x = -4i$, you might end up with the same confusion.