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$\begingroup$ Minor point: I notice quite a few elementary algebra books as well as some writers here taking the view that the n-th root of x is defined as x to the power 1/n.
The radical sign ' $√$ ' means we are taking the positive square root of given equation. if we simply say taking square roots on both sides, then we apply a ' $±$ ' before radical(' $√$ ') sign, as I said ' $√$ ' sign means positive square root, so in order to get negative one also we apply that ' $±$ ' sign.
You could say "the square roots of $49$ are $\pm 7$" and that would be fine; but otherwise saying "the square root of $49$" usually refers to what we write as $\sqrt{49}$. The $\sqrt{\ }$ symbol always refers to the positive root by default, so although $\sqrt{49}=7$ (which is positive) is 'the square root of $49$', $-\sqrt{49}=-7$ is another ...
$\begingroup$ The reason for preferring positive square roots of positive real numbers is practical. For instance, the distance formula in any finite number of dimensions is a positive square root - namely, for two points, the distance apart is the square root of the sum of the squares of the respective differences of coordinates.
The definite article and the singular in "the square root" is also important to imply the conventional definition of $\sqrt{}$. But more correctly he should say "the principal square root", because mathematically the expression "the square root" doesn't make sense, since there are two different roots in general.
I still carry the same confusion (which really bugs me) but I think the real confusion has to do with the square root and principal square root. I hope I can finally grasp the concept with the following two questions... Question 1: $\sqrt{12x^2} = \sqrt{4\cdot3x^2} = 2|x|\sqrt{3}$
If you force a choice of a single value out of the square root, it will break its algebraic properties. You may pick one and only one from this list: The square root is a function resulting in a single value. Care must be taken when doing algebra under the radical sign. The square root is a function resulting in perhaps many values.
[Try it out my multiplying it by itself.] It has no special notation beyond other complex numbers; in my discipline, at least, it comes up about half as often as the square root of 2 does --- that is, it isn't rare, but it arises only because of our prejudice for things which can be expressed using small integers. $\endgroup$ –
For example, $\\sqrt{4}$. I've asked a bunch of people and I get mixed answers all the time, as to whether it is $-2$ and $+2$ or just $+2$. How about if there's a negative in front of the square r...
We say that two complex numbers $\alpha$ and $\beta$ are conjugated if they both appear among the roots of a polynomial as above. Back to square roots. When the degree of the polynomial is $2$ , i.e. $$ P(X)=X^2+aX+b, $$ the roots are $\alpha_{1,2}=\frac12\left(-a\pm\sqrt{a^2-4b}\right)$ so that in fact one root is transformed into the other ...