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nth root

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This is the graph for [math]y=\sqrt{x}[/math]. It is a square root.
This is [math]y=\sqrt[3]{x}[/math]. It is a cube 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:


(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]


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]

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