Difference Between Two Radicals

[bergausstein]

can you tell the difference

$\displaystyle \sqrt{(1)^2}=1$$\displaystyle \sqrt{(-1)^2}=-1$

 

[MarkFL]

Use the definition:

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

Hence:

$$\sqrt{(-1)^2}=|-1|=1$$

 

[HallsofIvy]

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 $$x^2= 1$$.