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I have a question regarding this video

https://www.khanacademy.org/math/algebra/exponent-equations/exponent-properties-algebra/v/adding-and-simplifying-radicals

He said that $\sqrt {x^2}=|x|$ has to be the absolute value of $x$ because $x$ can be positive or negative, which makes sense but why is $\sqrt 4= 2$? Shoudln't it be $|2|$ too? Because $-2 \cdot -2 = 4$ and $2 \cdot 2 = 4$.

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1  
Indeed it is $|2|$. Therefore it is $2$. –  Michael Hardy Jan 1 '13 at 19:14
    
I'm changing the algebra to the algebra-precalculus tag, as the algebra is no longer being used. –  anorton Jan 2 '13 at 1:05

3 Answers 3

up vote 2 down vote accepted

$\left | 2 \right |=2$. Indeed $\sqrt{x^2}=\left | x \right |$ since $\sqrt{x^2}=x$ only for non-negative values of $x$. Consider for example this case:

$\sqrt{9}=\sqrt{(-3)^2}\neq -3$. But $\left | -3 \right |=3$ which is the correct answer.

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$\sqrt{9}=\sqrt{(-3)^2}\neq -3$ I am still a little bit confused, Wouldn't -3 and +3 both be correct results? I explained it to me the following way in this example $\sqrt{x^5}$ Because it is $x^5$ it has to be positive, because the square root of a negative number is undefined. But if we have $\sqrt{x^4}$ then x could be negative or positive. But maybe I am just a little bit confused. –  Maik Klein Jan 1 '13 at 17:40
    
$(-3)^2=9$ since$-3\times -3=9$. –  Basil R Jan 1 '13 at 17:54
    
@MaikKlein: What is your problem. Tell me explicitly. :-) –  Babak S. Jan 1 '13 at 18:02
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@Maik, when we write $\sqrt9$, we don't mean "any one of the two numbers whose square is $9$", we mean "the positive number whose square is $9$". That is the universally agreed-upon meaning of the $\sqrt{\cdot}$ symbol. See this previous question: Square roots — positive and negative. –  Rahul Jan 1 '13 at 18:07
    
Ah okay now it makes sense! That's the reason why we say $\sqrt{x^2} = | x$ | because the output of the function squareroot can only have 1 output , otherwise it wouldn't be a function. And the output is just defined to be positive, that why we have to force the x to be positive. Okay thank you very much –  Maik Klein Jan 1 '13 at 18:52

I'd make this a comment if I had the reputation, but as it stands, just think about what $|2|$ is.

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right, I am stupid thanks :X –  Maik Klein Jan 1 '13 at 17:25

Remember the definition of $|x|$: $$ |x| = \left\{ \begin{array}{ll} x & \quad x \geq 0 \\ -x & \quad x < 0 \end{array} \right. $$ So what is $|-2|$ according to this definition? It is $|-2|=-(-2)=2$

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