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Question Number 14811 by Nayon last updated on 04/Jun/17
why     (√x^2 )=∣x∣    ?
$${why}\:\:\:\:\:\sqrt{{x}^{\mathrm{2}} }=\mid{x}\mid\:\:\:\:?\: \\ $$
Answered by mrW1 last updated on 04/Jun/17
if x≥0, (√x^2 )=x=∣x∣  if x<0, (√x^2 )=(√((−x)^2 ))=(−x)=∣x∣  ⇒(√x^2 )=∣x∣ (always valid)
$${if}\:{x}\geqslant\mathrm{0},\:\sqrt{{x}^{\mathrm{2}} }={x}=\mid{x}\mid \\ $$$${if}\:{x}<\mathrm{0},\:\sqrt{{x}^{\mathrm{2}} }=\sqrt{\left(−{x}\right)^{\mathrm{2}} }=\left(−{x}\right)=\mid{x}\mid \\ $$$$\Rightarrow\sqrt{{x}^{\mathrm{2}} }=\mid{x}\mid\:\left({always}\:{valid}\right) \\ $$

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