Stepenovanje korena

Pravilo glasi:

$$\bigl(\sqrt[n]{a}\bigr)^{m} = \bigl(a^{\tfrac{1}{n}}\bigr)^{m} = a^{\tfrac{m}{n}}$$

Jednako važna je i obrnuta formulа, korenovanje stepena:

$$\sqrt[n]{a^{m}} = \bigl(a^{m}\bigr)^{\tfrac{1}{n}} = a^{\tfrac{m}{n}}$$

Dakle, podizanje korena na stepen ili izvlačenje korena iz stepena daje isti eksponentni izraz $a^{m/n}$.

Primeri

$$(\sqrt{a})^2 = a^{\tfrac{1}{2} \cdot 2} = a^1 = a$$ $$\bigl(\sqrt[3]{x}\bigr)^4 = x^{\tfrac{4}{3}}$$ $$\sqrt[5]{m^3} = m^{\tfrac{3}{5}}$$ $$\sqrt{\bigl(x^2 y^3\bigr)} = (x^2 y^3)^{\tfrac{1}{2}} = x^{2\cdot\tfrac{1}{2}} \, y^{3\cdot\tfrac{1}{2}} = x^1 \, y^{\tfrac{3}{2}} = x \, y^{\tfrac{3}{2}}$$

5. Koren od korena (složen koren):

$$\sqrt[m]{\sqrt[n]{a}} = \bigl(a^{\tfrac{1}{n}}\bigr)^{\tfrac{1}{m}} = a^{\tfrac{1}{n}\cdot \tfrac{1}{m}} = a^{\tfrac{1}{mn}}$$