nth root

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

This is [math]\displaystyle{ 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:

[math]\displaystyle{ k^n=r }[/math]

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

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

For example, [math]\displaystyle{ \sqrt[3]{8} = 2 }[/math] because [math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ \sqrt{ab} = \sqrt{a} \times \sqrt{b} }[/math]. The quotient property of a radical expression is the statement [math]\displaystyle{ \sqrt{\tfrac{a}{b}} = \tfrac{\sqrt{a}}{\sqrt{b}} }[/math]., b != 0.

Simplifying

This is an example of how to simplify a radical.

[math]\displaystyle{ \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.

[math]\displaystyle{ 2\sqrt{2} + 1\sqrt{2} = 3\sqrt{2} }[/math]

[math]\displaystyle{ 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]\displaystyle{ \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]

Nth Root Media

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  The four 4th roots of −1, none of which are real

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  The three 3rd roots of −1, one of which is a negative real

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  The graph of y=x{\displaystyle y={\sqrt {x}}}

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  The graph of y=∛x

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  Pascal's triangle showing P(4,1) = 4.

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  The square roots of i

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  The third roots of unity

Related pages

• Geometric mean
• Rationalization (mathematics)
• Square number