Why is $\sqrt{X}\times\sqrt{X}=X$?

Question

Today I was solving the limit $(\ln(x))/(2*(x^{1/2})$ but then faced the step after applying the derivation that ended up with $(1/x)/(1/x^{1/2})$ and the result of that was $1/x^{1/2}$. When I asked a friend to explain why he said it's because the $X$ is replaced with $\sqrt X * \sqrt X$. So does that means that $\sqrt X * \sqrt X = X$? if so can someone explain why?

Answer 1

That is the definition of square root. A square root of a real number $x\ge0$ is another real number $a$ such that $$ a^2=x. $$ If $x>0$ it has two square roots, denoted as $$ \pm\sqrt x, $$ where it is understood that $\sqrt x>0$. Then $$ \sqrt x\,\sqrt x=(\sqrt x)^2=x. $$