Działania na pierwiastkach

Jeżeli $a \ge 0, b \ge 0, n,m \in N \backslash \{0, 1\}$, to:

$\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$

$\sqrt[m]{ \sqrt[n]{a} } = \sqrt[mn]{a}$

$(\sqrt[n]{a})^m = \sqrt[n]{a^m}$

$(\sqrt[n]{a})^n = a$

$a \cdot \sqrt[n]{b} = \sqrt[n]{a^nb}$

$\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$, dla $b \gt 0$

Przykłady
$\sqrt{2} \cdot \sqrt{5} = \sqrt{10}$
$2\sqrt{3} \cdot \sqrt{2} = 2\sqrt{6}$
$\sqrt{3} \cdot \sqrt{3} = \sqrt{9} = 3$
$\sqrt{2} \cdot \sqrt{6} = \sqrt{12} = 2\sqrt{3}$
$\sqrt[3]{2} \cdot \sqrt[3]{4} = \sqrt[3]{8} = 2$
$\sqrt[3]{3} \cdot \sqrt[3]{5} = \sqrt[3]{15}$

$\sqrt[3]{ \sqrt[2]{8} } = \sqrt[6]{8}$
$\sqrt{ \sqrt{10} } = \sqrt[4]{10}$

$(\sqrt[3]{4})^2 = \sqrt[3]{4^2} = \sqrt[3]{16}$
$(\sqrt{3})^2 = \sqrt{3^2} = \sqrt{9} = 3$
$(\sqrt{10})^3 = \sqrt{10^3} = \sqrt{1000}$

$3 \cdot \sqrt{2} = \sqrt{3^2 \cdot 2} = \sqrt{18}$
$2 \cdot \sqrt[3]{5} = \sqrt[3]{2^3 \cdot 5} = \sqrt[3]{40}$

$\sqrt{\frac{4}{9}} = \frac{\sqrt4}{\sqrt9} = \frac{2}{3}$
$\sqrt[3]{\frac{3}{8}} = \frac{\sqrt[3]3}{\sqrt[3]8} = \frac{\sqrt[3]3}{2}$