# Operations with Roots

Real numbers: $$a$$, $$b$$, $$p$$
Natural numbers: $$n$$, $$m$$
1. The $$n$$th root of a number $$a$$ is called a number $$b,$$ the $$n$$th power of which equals $$a.$$ Here $$a$$ and $$b$$ are real numbers, $$n$$ is a natural number $$\left({n \ge 2}\right).$$
$$\sqrt[\large n\normalsize]{a} = b,\;\;{b^n} = a$$
2. The principal or arithmetic $$n$$th root of a nonnegative number $$a$$ is called a nonnegative number $$b,$$ the $$n$$th power of which equals $$a.$$ If $$a = 0,$$ the principal $$n$$th root is also zero:
$$\sqrt[\large n\normalsize]{0} = {0^{1/n}} = 0$$
3. If $$a \lt 0,$$ the $$n$$th root of a number $$a$$ is defined only for odd indices n.
4. The square root of a number $$a$$ $$\left(a \ge 0\right)$$ is usually denoted as $$\sqrt a$$.
5. $$N$$th root of a product $$\sqrt[\large n\normalsize]{{ab}} = \sqrt[\large n\normalsize]{a}\sqrt[\large n\normalsize]{b}$$
6. Multiplication of roots with different bases and different indices
$$\sqrt[\large n\normalsize]{a}\sqrt[\large m\normalsize]{b} = \sqrt[{\large nm\normalsize}]{{{a^m}{b^n}}}$$
7. $$N$$th root of a quotient
$$\sqrt[\large n\normalsize]{{\large\frac{a}{b}}}\normalsize = \large\frac{{\sqrt[n]{a}}}{{\sqrt[n]{b}}}\normalsize\;\;\left( {b \ne 0} \right)$$
8. Division of roots with different bases and different indices
$${\large\frac{{\sqrt[n]{a}}}{{\sqrt[m]{b}}}\normalsize} =$$ $${\large\frac{{\sqrt[{nm}]{{{a^m}}}}}{{\sqrt[{nm}]{{{b^n}}}}}\normalsize} =$$ $${\large\sqrt[{nm}]{{\frac{{{a^m}}}{{{b^n}}}}}\normalsize}\;$$ $$\left( {b \ne 0} \right)$$
9. Raising a root to a power $${\left( {\sqrt[\large n\normalsize]{a}} \right)^m} = \sqrt[\large n\normalsize]{{{a^m}}}$$
10. $${\left( {\sqrt[\large n\normalsize]{a}} \right)^n} = a$$
11. $$N$$th root of a power $$\sqrt[\large n\normalsize]{{{a^m}}} = {a^{m/n}}$$
12. $$\sqrt[\large n\normalsize]{{{a^m}}} = \sqrt[{\large np\normalsize}]{{{a^{mp}}}}$$
13. $${\left( {\sqrt[\large n\normalsize]{{{a^m}}}} \right)^p} = \sqrt[\large n\normalsize]{{{a^{mp}}}}$$
14. Root of a root $$\sqrt[\large m\normalsize]{{\sqrt[\large n\normalsize]{a}}} = \sqrt[{\large mn\normalsize}]{a}$$
15. Reciprocal of a root
$${\large\frac{1}{{\sqrt[n]{a}}}\normalsize} = {\large\frac{{\sqrt[n]{{{a^{n – 1}}}}}}{a}\normalsize}\;$$ $$\left( {a \ne 0} \right)$$
16. $$\sqrt {a \pm \sqrt b } =$$ $$\sqrt {\large\frac{{a + \sqrt {{a^2} – b} }}{2}\normalsize} \pm \sqrt {\large\frac{{a – \sqrt {{a^2} – b} }}{2}\normalsize} \;$$ $$\left( {b \ge 0,a \ge \sqrt b } \right)$$
$$\sqrt {a + \sqrt b } \pm \sqrt {a – \sqrt b } =$$ $$2\sqrt {\large\frac{{a \pm \sqrt {{a^2} – b} }}{2}\normalsize} \;$$ $$\left( {b \ge 0,a \ge \sqrt b } \right)$$
$${\large\frac{1}{{\sqrt a \pm \sqrt b }}\normalsize} = {\large\frac{{\sqrt a \mp \sqrt b }}{{a – b}}\normalsize}\;$$ $$\left( {a \ne b} \right)$$