Positive and negative square roots

Question

If the number $13$ is squared it gives $169$. Then if we take the square root of $169$; $\sqrt{169}$ it gives $13$ and $-13$. Why is this so if we know that $13$ was positive and it was multiplied by itself and produced $169$.

Answer 1

Always note that $\sqrt{x}$ always gives a positive number.

So in your example: $$\sqrt{169} = +13$$

But, If an equation is given like: $$x^2 = 169$$ then, $$x = +13 (or) -13$$

Answer 2

because squaring is not injective. Specifically, $(13)^2 = 169$ but $(-13)^2=169$ also. Square root is supposed to invert a square but in fact the inverse is not a function.

If you're talking in particular about what you put into a calculator, the calculator doesn't "remember" what the input value was. Internally, it stores whatever the result was. So your calculator doesn't know the difference between the result of $(-13)^2$ and $(13)^2$. There's no real reason that it couldn't, they just don't.

Answer 3

Actual definition of square root function is $\sqrt{X^2}=|X|$

All square roots of $X^2$ is indeed $-\sqrt{X}$ and $\sqrt{X^2}$.

Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive a, the principal square root can also be written in exponent notation, as $a^{\frac{1}{2}}$. Source:Square root