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# Basic properties of radicals and examples

Get to know the roots of math with their basic properties and review some examples of using that.

First of all, there is a table of basic properties of radicals. With these properties you can solve many tasks included nth roots.

\begin{array}{|c|c|} \hline \sqrt{a}\cdot\sqrt{b}=\sqrt{ab} & \frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}} \\\hline \sqrt{a^n}=(\sqrt{a})^n & (\sqrt a)^2=a \\\hline a\sqrt b =\sqrt{a^2\cdot b} & \sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a} \\\hline \end{array}

## Solved examples

Solve this expression $2\sqrt{25}-3\sqrt{125}+\sqrt{-216}$

$$2\sqrt{25}-3\sqrt{125}+\sqrt{-216}=2\cdot 5 - 3\cdot 5 - 6 = -11$$

Simplify this expression $(2\sqrt{3}-1)(2-\sqrt{3})-5\sqrt{3}$

$$(2\sqrt{3}-1)(2-\sqrt{3})-5\sqrt{3}=4\sqrt{3}-6-2+\sqrt{3}-5\sqrt{3}=-8$$

Simplify this expression $(1+3\sqrt{3})^2-(3+\sqrt{3})^2$

$$(1+3\sqrt{3})^2-(3+\sqrt{3})^2=1+6\sqrt{3}+27-9-6\sqrt{3}-3=16$$

Simplify this expression $2\sqrt{75}+5\sqrt{27}-3\sqrt{48}$

$$2\sqrt{75}+5\sqrt{27}-3\sqrt{48}=2\sqrt{3\cdot 25}+5\sqrt{3\cdot 9}-3\sqrt{3\cdot 16}=10\sqrt{3}+15\sqrt{3}-12\sqrt{3}=13\sqrt{3}$$

Simplify this expression $\displaystyle\frac{2\sqrt{32}+\sqrt{128}}{5\sqrt{200}-\sqrt{50}}$

$$\frac{2\sqrt{32}+\sqrt{128}}{5\sqrt{200}-\sqrt{50}}=\frac{2\sqrt{2\cdot 16}+\sqrt{2\cdot 64}}{5\sqrt{2\cdot 100}-\sqrt{2\cdot 25}}=\frac{8\sqrt{2}+8\sqrt{2}}{50\sqrt{2}-5\sqrt{2}}=\frac{16\sqrt{2}}{45\sqrt{2}}=\frac{16}{45}$$