MHB Difference Between Two Radicals

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The discussion clarifies the distinction between the square roots of positive and negative numbers, emphasizing that both (1)^2 and (-1)^2 equal 1. It highlights that while the square root of 1 is defined as the positive value, the expression √((-1)^2) is incorrectly stated as -1. The correct interpretation is that √((-1)^2) equals 1, aligning with the definition of absolute value. Thus, the key takeaway is that the square root function returns the principal (positive) root. Understanding this distinction is essential for accurate mathematical reasoning.
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can you tell the difference

$\displaystyle \sqrt{(1)^2}=1$$\displaystyle \sqrt{(-1)^2}=-1$
 
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Use the definition:

$$|x|\equiv\sqrt{x^2}$$

Hence:

$$\sqrt{(-1)^2}=|-1|=1$$
 
bergausstein said:
can you tell the difference

$\displaystyle \sqrt{(1)^2}=1$$\displaystyle \sqrt{(-1)^2}=-1$
The major difference is that the first one is right and the second one is wrong! Both (1)^2 and (-1)^2 are equal to 1 so both of those is \sqrt{1}. And \sqrt{a} is defined as the positive number, x, such that x^2= 1.
 

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