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*n*th root

An ** n-th root** of a number

*r*is a number which, if

*n*copies are multiplied together, makes

*r*. It is also called a

**radical**or a

**radical expression**. It is a number

*k*for which the following equation is true:

- [math]k^n=r[/math]

(for the meaning of [math]k^n[/math], see Exponentiation.)

We write the nth root of *r* as [math]\sqrt[n]{r}[/math].^{[1]} If *n* is 2, then the radical expression is a **square root**. If it is 3, it is a **cube root**.^{[2]}^{[3]} Other values of n are referred to using ordinal numbers, such as *fourth root* and *tenth root*.

For example, [math]\sqrt[3]{8} = 2[/math] because [math]2^3 = 8[/math]. The 8 in that example is called the **radicand**, the 3 is called the **index**, and the check-shaped part is called the **radical symbol** or **radical sign**.

Roots and powers can be changed as shown in [math]\sqrt[b]{x^a} = x^\frac{a}{b} = (\sqrt[b]{x})^a = (x^a)^\frac{1}{b}[/math].

The **product property** of a radical expression is the statement that [math]\sqrt{ab} = \sqrt{a} \times \sqrt{b}[/math]. The **quotient property** of a radical expression is the statement [math]\sqrt{\tfrac{a}{b}} = \tfrac{\sqrt{a}}{\sqrt{b}}[/math].^{[3]}

## Simplifying

This is an example of how to simplify a radical.

- [math]\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}[/math]

If two radicals are the same, they can be combined. This is when both of the indexes and radicands are the same.^{[4]}

- [math]2\sqrt{2} + 1\sqrt{2} = 3\sqrt{2}[/math]
- [math]2\sqrt[3]{7} - 6\sqrt[3]{7} = -4\sqrt[3]{7}[/math]

This is how to find the perfect square and rationalize the denominator.

- [math]\frac{8x}{\sqrt{x}^3} = \frac{8\cancel{x}}{\cancel{x}\sqrt{x}} = \frac{8}{\sqrt{x}} = \frac{8}{\sqrt{x}} \times \frac{\sqrt{x}}{\sqrt{x}} = \frac{8\sqrt{x}}{\sqrt{x}^2} = \frac{8\sqrt{x}}{x}[/math]

## Related pages

## References

- ↑ "List of Arithmetic and Common Math Symbols" (in en-US). 2020-03-17. https://mathvault.ca/hub/higher-math/math-symbols/common-math-symbols/.
- ↑ Weisstein, Eric W.. "nth Root" (in en). https://mathworld.wolfram.com/nthRoot.html.
- ↑
^{3.0}^{3.1}"nth Roots". https://www.mathsisfun.com/numbers/nth-root.html. - ↑ "Add and Subtract Radicals". https://mathbitsnotebook.com/Algebra1/Radicals/RADAddSubtract.html.