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?
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2you ask why $\sqrt{xy}=\sqrt{x}\cdot \sqrt{y}$? Just square both sides of the equation (this is allowed because both sides are positive). – Yanko Nov 23 '19 at 16:39
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1for $A,B\geq 0$ – AlvinL Nov 23 '19 at 17:02
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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 $y=x^{log_x y}$
$$ (xy)^a=(x\times x^{\log_xy})^a \\=x^{a+a\log_x y} \\ = x^a y^a $$
WW1
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