Algebra Roots Formulas

Algebra Roots Formulas
Bases: a, b
Powers (rational numbers): n ,m
a , b ≥ 0 for even roots (n =2k, k $\in$ N)

1. $\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}$

2. $\sqrt[n]{a} \sqrt[m]{b}= \sqrt[nm]{a^{m}b^{n}}$

3. $\sqrt[n]{\frac{a}{b}}= \frac{\sqrt[n]{a}}{\sqrt[n]{b}} , b \neq 0$

4. $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} =\frac{\sqrt[nm]{a^{m}}}{\sqrt[nm]{b^{n}}}= \sqrt[nm]{\frac{a^{m}}{b^{n}}}, b \neq 0.$

5. $(\sqrt[n]{a^{m}})^{}p = \sqrt[n]{a^{mp}}$

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

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

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

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

10. $\frac{1}{\sqrt[n]{a}} =\frac{\sqrt[n]{a^{n-1}}}{a},a \neq 0.$

11. $\sqrt{a\pm \sqrt{b}} = \sqrt{\frac{a+\sqrt{a^{2}-b}}{2}} \pm \sqrt{\frac{a-\sqrt{a^{2}-b}}{2}}$

12. $\frac{1}{\sqrt{a}\pm \sqrt{b}} = \frac{\sqrt{a}\pm \sqrt{b}}{a-b}$