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### Significance of $\sqrt[n]{a^n}$?!

There is a formula given in my module: $$\sqrt[n]{a^n} = \begin{cases} \, a &\text{ if n is odd } \\ |a| &\text{ if n is even } \end{cases}$$ I don't really understand the ...
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### Why is $\sqrt{4} = 2$ and Not $\pm 2$? [duplicate]

I've always been told that if $\ x^2 = 4,$ $=>x = \pm2$ But recently, Prof. mentioned that if $x = \sqrt{4}$, Then $x = +2(only)$ I am very skeptical about this because they both mean the ...
383 views

### Why does the square root of a square involve the plus-minus sign?

If $\sqrt{x^2}$ can be simplified as follows: $\sqrt{x^2} = (x^2)^\frac{1}{2} = x^{\frac{2}{1}\times\frac{1}{2}} =x^\frac{2}{2} = x^1 = x$ Then why would $\sqrt{x^2} = \pm x$?
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### A question about eliminating square roots [duplicate]

If $\sqrt{x^2} = \pm x$, then why does $\sqrt{(x+2)^2} = x+2$ and not $\pm (x+2)$? This is driving me crazy, so feel free to elucidate. Thanks! ---EDIT--- I'm not sure how the other questions' ...
### Why does Wolfram Alpha say that $\sqrt{1}=-1$? [duplicate]
Why does Wolfram Alpha say that $\sqrt{1}=-1$? Is this a mistake or what? Can anyone help? Thanks in advance.