nth root

File:Squareroot-0-9-metapost.svg

This is the graph for <math>y=\sqrt{x}</math>. It is a square root.

File:Cube-root function.svg

This is <math>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>k^n=r</math>

(for the meaning of <math>k^n</math>, see Exponentiation.)

We write the nth root of r as <math>\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>\sqrt[3]{8} = 2</math> because <math>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>\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>\sqrt{ab} = \sqrt{a} \times \sqrt{b}</math>. The quotient property of a radical expression is the statement <math>\sqrt{\tfrac{a}{b}} = \tfrac{\sqrt{a}}{\sqrt{b}}</math>., b != 0.

Simplifying

This is an example of how to simplify a radical.

<math>\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>2\sqrt{2} + 1\sqrt{2} = 3\sqrt{2}</math>

<math>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>\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>

Related pages

• Geometric mean
• Rationalization (mathematics)
• Square number