Why in these types of questions(the image) I need to multiply the dominator by itself and by the nominator. and why I can't just multiply it just by itself.

Answer:
[tex]\frac{\sqrt{5x^{2} } }{\sqrt{3x^{3} } } =\sqrt{\frac{5}{3x} }[/tex]
Step-by-step explanation:
Simply because
[tex]A=\frac{5}{3} = \frac{5}{3}*\frac{3}{3} = \frac{15}{9} =\frac{5}{3}[/tex]
So its value remains Constant , It's simply as if you're multiplying by 1 which doesn't change anything at all
but when doing this
[tex]A=\frac{5}{3} = \frac{5}{3}*\frac{5}{3} = \frac{25}{9} \neq \frac{5}{3}\\[/tex]
You changed its value
[tex]A[/tex] now became [tex]A^{2}[/tex]
Hope you understand it