Wprowadźmy oznaczenia:

$x=\frac{m}{n}\text{i}y=\frac{p}{q},m\in \mathbb{Z},p\in \mathbb{Z},n\in {\mathbb{Z}}^{+},q\in {\mathbb{Z}}^{+}$  

Udowodnijmy podane równości:

$a\text{)}{a}^x:a^y=a^{x-y}$  

$a^x:a^y=a^{\frac{m}{n}}:a^{\frac{p}{q}}=a^{\frac{m}{n}}\cdot \frac{1}{a^{\frac{p}{q}}}=a^{\frac{m}{n}}\cdot {\left(\frac{1}{a}\right)}^{\frac{p}{q}}=$  

$=a^{\frac{m}{n}}\cdot {\left(a^{-1}\right)}^{\frac{p}{q}}=a^{\frac{m}{n}}\cdot a^{-\frac{p}{q}}=a^{\frac{m}{n}-\frac{p}{q}}=a^{x-y}$  

c.k.d.

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$b\text{)}{\left(\frac{a}{b}\right)}^x={\left(\frac{a}{b}\right)}^{\frac{m}{n}}={\left(a\cdot \frac{1}{b}\right)}^{\frac{m}{n}}=a^{\frac{m}{n}}\cdot {\left(\frac{1}{b}\right)}^{\frac{m}{n}}=$  

$=a^{\frac{m}{n}}\cdot \frac{1}{b^{\frac{m}{n}}}=\frac{a^{\frac{m}{n}}}{b^{\frac{m}{n}}}=\frac{a^x}{b^x}$  

c.k.d.