Działania na pierwiastkach

Jeżeli a ≥ 0, b ≥ 0, n ∈ N\{0, 1}, to:

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

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

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

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

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

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

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

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

${(\sqrt[3]{4})}^{2} = \sqrt[3]{4^{2}} = \sqrt[3]{16}$
${(\sqrt{3})}^{2} = \sqrt{3^{2}} = \sqrt{9} = 3$
${(\sqrt{10})}^{3} = \sqrt{1 0^{3}} = \sqrt{1000}$

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

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