Soient \(a \in {\mathbb{R}^ + }\), \(b \in {\mathbb{R}^ + }\), \(n \in \mathbb{Q}\) et \(m \in \mathbb{Q}\).

1. \(\sqrt[n]{a} = {a^{\frac{1}{n}}}\)

2. \(\sqrt[n]{{ab}} = \sqrt[n]{a}\sqrt[n]{b}\)

3. \(\sqrt[n]{a}\sqrt[m]{b} = \sqrt[{nm}]{{ab}}\)

4. \(\sqrt[n]{{\frac{a}{b}}} = \frac{{\sqrt[n]{a}}}{{\sqrt[n]{b}}}\) avec \(b \ne 0\)

5. \(\frac{{\sqrt[n]{a}}}{{\sqrt[m]{b}}} = \frac{{\sqrt[{n,m}]{{{a^m}}}}}{{\sqrt[{m.n}]{{{b^n}}}}}\) \( = \sqrt[{n.m}]{{\frac{{{a^m}}}{{{b^n}}}}}\) avec \(b \ne 0\)

6. \({\left( {\sqrt[n]{{{a^m}}}} \right)^p} = \sqrt[n]{{{a^{m.p}}}}\)

7. \({\left( {\sqrt[n]{a}} \right)^n} = a\)

8. \(\sqrt[n]{{{a^m}}} = \sqrt[{n.p}]{{{a^{m.p}}}}\)

9. \(\sqrt[n]{{{a^m}}} = {a^{\frac{m}{n}}}\)

10. \(\sqrt[m]{{\sqrt[n]{a}}} = \sqrt[{m.n}]{a}\)

11. \({\left( {\sqrt[n]{a}} \right)^m} = \sqrt[n]{{{a^m}}}\)

12. \(\frac{1}{{\sqrt[n]{a}}} = \frac{{\sqrt[n]{{{a^{n - 1}}}}}}{a}\)

13. \(\sqrt {a \pm \sqrt b } = \) \(\sqrt {\frac{{a + \sqrt {{a^2} - b} }}{2}} \pm \) \(\sqrt {\frac{{a - \sqrt {{a^2} - b} }}{2}} \)

14. \(\frac{1}{{\sqrt a \pm \sqrt b }} = \) \(\frac{{\sqrt a \pm \sqrt b }}{{a - b}}\)