Values of square roots Good-morning Math Exchange (and good evening to some!)
I have a very basic question that is confusing me. 
At school I was told that $\sqrt {a^2} = \pm a$
However, does this mean that $\sqrt {a^2} = +2$ *and*$-2$ or does it mean:
$\sqrt {a^2} = +2$ *OR*$-2$
Is it wrong to say 'and'? What are the implications of choosing 'and'/'or'
Any help would be greatly appreciated. Thanks in advance and enjoy the rest of your day :)
 A: But the truth being that the squarte root function is always associated with absolute value function. That is
$\sqrt {a^2} = \vert a \vert $ 
This is very very important to remember and be careful while using the $ \sqrt{} 
$ .
Note that if it is  $ {(a^{2}})^{\frac{1}{2}} $ then we get answer as $\pm a $ but note that  $\sqrt {a^2} = \vert a \vert $
Absoulte value is associated as OR
$\vert a \vert$ = $a$ OR $-a$ 
And is not used as a variable cant have two values at the same time !!
A: Even though quite a bit has been said already, i wanted to add something. 
The numbers which you normally use in school (-1, $\frac{2}{3}$, $ \pi$, etcetera) are called the real numbers. The set of real numbers is denoted by $\mathbb{R}$.
Now the square root of any number $b$ is normally considered to be any number $x$ that satisfies $x^2 = b$, or equivalently $x^2 - b = 0$.
As you pointed out, there are normally two solutions to this, so two values for $x$ will do the trick. However, working in $\mathbb{R}$ this situation is remedied by adopting the convention that the square root of $b$ will be the positive number $x$ that satisfies $x^2 - b=0$. So indeed, when $b= a^2$, we get
$$
\sqrt{a^2} = |a|.
$$
So with this convention, the solutions to $x^2 - b=0$ become $x=\sqrt{b}$ and $x = -\sqrt{b}$. It is very important to note that this is merely a convention.
Even more: there are more sets of numbers we could work in, where this trick will not work! If we pass from the real numbers $\mathbb{R}$ to the so called complex numbers, denoted $\mathbb{C}$ (check wikipedia), we lose this! In this set of numbers, the notion of a positive number does not make sense, and it is in fact impossible to define a square root function in a nice way on the whole of $\mathbb{C}$ (if you want to know more about this, ask google).
In general there are many more things that i call "sets of numbers" now, in mathematics they are called "fields". In all of them, the square root notion makes sense, as in solving the solution to $x^2 - b=0$. However, the nice $\sqrt{{}}$ function as we have it in $\mathbb{R}$ is rarely found in other fields.
Hope this context was interesting to you.
A: $\sqrt{\mathrm{a}^{2}}=|a|$= +a OR -a.Remember here OR is used.AND is not used as a function cannot have two values at same time.
A: The usual convention is that the square root sign gives the positive square root. When the negative/both is/are required, use the minus/plusminus sign.
