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Is $\sqrt{(-10)^2}$ equivalent to $-10$ or $10$, or is it equivalent to only one among the two?

Since $\sqrt{(-10)^2} = \sqrt{100}$ and $\sqrt{100} =$ $-10$ or $10$. Using this solution, it can be equivalent to either the two answers.

But using this solution:

$\sqrt{(-10)^2} = -10^{\frac{2}{2}}$

$-10^{\frac{2}{2}} = -10$

It has only 1 answer.

AYA
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  • Well square root of a number is never negative. Just look at the curve $y=\sqrt{x}$. It's in the first quadrant $\forall x,y$. The statement would be false. – sai-kartik Apr 14 '20 at 06:06
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    The answer is $10$ and only $10$ because the radical sign means, by definition, the positive square root. – John Douma Apr 14 '20 at 06:06
  • So many people have the same misconception. I don't think the concept is being taught well around the world. – Toby Mak Apr 14 '20 at 06:30
  • The context or definition I am using is this. "The number $a$ is the square root of $b$ in the expression $a^2 = b$. This means that if you multiply $a$ by itself, or $a$ by $a$, you will get $b$. From this, we can say that a number has a negative square root. – AYA Apr 14 '20 at 06:34

2 Answers2

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We know that the function $y=x^2$ isn't invertible in all the domain, so in order to find the inversr, we can consider: $$f^{-1}: R^+\rightarrow R^+$$ So, the correct answer is $\sqrt{10^2}=10$ and you can check this using a graph calculator.

Note that evn if you are using complex numbers, then you obtain again $\sqrt{(-10)^2}=10$ and not $\pm10i$ because: $$(\pm10i)^2=-100$$

Matteo
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  • Thank you for your answer. If the context will include complex numbers, can $\sqrt{(-10)^2} = \sqrt{100}$ =$ $-10$ or $10$? – AYA Apr 14 '20 at 06:22
  • @MKA: please see my edits. – Matteo Apr 14 '20 at 06:29
  • Thanks for the updated solution! My question now is "Is it possible to use another context to say that the square root of a number can be equivalent to a negative number?" – AYA Apr 14 '20 at 06:33
  • Yes, if you choose the inteval of bijectivity of $y=x^2$ in $(-\infty,0]$, you can define the square root as follows: $$\sqrt{x}: R^+\rightarrow R^-$$ – Matteo Apr 14 '20 at 06:36
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$$\sqrt x$$ is a function of $x$. As such, it can only take one value per $x$.