Is$\frac{\sqrt{a}}{\sqrt{b}}$ the same as $\sqrt{\frac{a}{b}}$?

Solution 1:

Let us define the functions: $$ f(a,b)=\frac{\sqrt{a}}{\sqrt{b}}\quad\,\,\text{and}\,\,\quad g(a,b)=\sqrt{\frac{a}{b}}. $$ Then $f$ and $g$ AGREE on the intersection of their domains. However, they have different domains:

$$ \mathrm{Dom}(f)=\{(a,b): a\ge 0,\,\,b>0\}, $$ while $$ \mathrm{Dom}(g)=\{(a,b): a\ge 0,\,\,b>0\}\cup\{(a,b): a\le 0,\,\,b<0\}. $$

Strictly speaking, in order for two functions to be equal they need to have the same domain (and they same values for each element of their domain.) Hence, strictly speaking, these functions are not the equal.

Solution 2:

Actually, you are right- as functions of two real variables, they have different domains. For example take $a=-1,b=-1$