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 $\sqrt[n]{r}$. 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 $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$., 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.

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

Related pages

• Geometric mean
• Rationalization (mathematics)
• Square number