# nth root

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

An n-th root of a number r is a number which, if multiplied by itself n times, makes r. It is also called a radical or a radical expression. You could say that it is a number k for which this equation is true:

${\displaystyle k^{n}=r}$

(for meaning of ${\displaystyle k^{n}}$, read exponentiation.)

We write it like this: ${\displaystyle {\sqrt[{n}]{r}}}$. If n is 2, then the radical expression is a square root. If it is 3, it is a cube root.

For example, ${\displaystyle {\sqrt[{3}]{8}}=2}$ because ${\displaystyle 2^{3}=8}$. 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 ${\displaystyle {\sqrt[{b}]{x^{a}}}=x^{\frac {a}{b}}=({\sqrt[{b}]{x}})^{a}=(x^{a})^{\frac {1}{b}}}$.

The product property of a radical expression is shown in ${\displaystyle {\sqrt {ab}}={\sqrt {a}}\times {\sqrt {b}}}$.

The quotient property of a radical expression is shown in ${\displaystyle {\sqrt {\frac {a}{b}}}={\frac {\sqrt {a}}{\sqrt {b}}}}$.

## Simplifying

This is an example of how to simplify a radical.

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

If two radicals are the same, they can be combined. This is when both of the indexes and radicands are the same.[1]

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

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

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