kidzsearch.com > wiki   Explore:web images videos games

# nth root

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

$k^n=r$

(for the meaning of $k^n$, see Exponentiation.)

We write the nth root of r as $\sqrt[n]{r}$.[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, $\sqrt[3]{8} = 2$ because $2^3 = 8$. 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 $\sqrt[b]{x^a} = x^\frac{a}{b} = (\sqrt[b]{x})^a = (x^a)^\frac{1}{b}$.

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

## Simplifying

This is an example of how to simplify a radical.

$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$

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

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

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

$\frac{8x}{\sqrt{x}^3} = \frac{8\cancel{x}}{\cancel{x}\sqrt{x}} = \frac{8}{\sqrt{x}} = \frac{8}{\sqrt{x}} \times \frac{\sqrt{x}}{\sqrt{x}} = \frac{8\sqrt{x}}{\sqrt{x}^2} = \frac{8\sqrt{x}}{x}$