What's the proof that $\sqrt {AB} = \sqrt A \times \sqrt B$

Question

Sorry for simple questions, just trying to understand the basics. I suppose since $\sqrt x = x^{\frac12}$ the proof must be the same as: $$(AB)^x =A^x \times B^x $$ But then how do you prove that?

Answer 1

The minimal set of axioms to define exponentiation are .. $$ x^1=x \\ x^ax^b=x^{a+b}$$

from which you can prove the rule $(x^a)^b=x^{ab}$

From here we can prove your result using $y=x^{log_x y}$

$$ (xy)^a=(x\times x^{\log_xy})^a \\=x^{a+a\log_x y} \\ = x^a y^a $$