# nth root

function

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 ${\sqrt[{n}]{r}}$ . 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, ${\sqrt[{3}]{8}}=2$ 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 ${\sqrt {\tfrac {a}{b}}}={\tfrac {\sqrt {a}}{\sqrt {b}}}$ ., b != 0.

## 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.

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

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

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