# Roots and Power

Description of the calculation with Roots, Power and Exponents with examples

## Description

This page describes a general relationship between roots and powers in mathematical terms.

First to the power; they can be considered as shorthand of multiplication.

The expression $$a^{4}$$ stands for $$a · a · a · a$$.

In the expression $$a^n$$ we call $$a$$ the basis and $$n$$ the exponent.

For a negative exponen $$a^{-n}$$ you can also write $$1/a^{n}$$.

A general root for natural numbers is also defined by the exponent

In $$\sqrt[n]{a}$$ we call $$a$$ the radicand and $$n$$ again the exponent

It is $$\sqrt{8}=2$$ or $$\sqrt{16}=4$$, where without specifying the exponent, the $$2$$ is assumed as an exponentwird.

If $$\sqrt[n]{a}=b$$, then $$b^{n}=a$$.

## List of rules

The following list shows some rules that simplify the process of converting and calculating formulas

• $$a^{n}·a^{m} = a^{n + m}$$

• $$\frac{a^{n}}{a^{m}} = a^{n-m}$$

• $$a^{n}·b^{n}=(ab)^{n}$$

• $$\sqrt[n]{a^{n}}=(\sqrt[n]{a})^n=a$$

• $$\displaystyle\frac{a^n}{b^n}=(\frac{a}{b})^n$$

• $$(a^n)^m=a^{nm}$$

• $$a^0=1$$

• $$\sqrt[n]{1}=1$$

• $$\sqrt[n]{\sqrt[m]{a}}=\sqrt[n-m]{a}$$

• $$\displaystyle\frac{a}{\sqrt{a}}= \sqrt{a}$$

• $$\displaystyle\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$$

• $$\sqrt[n]{a}·\sqrt[n]{b}=\sqrt[n]{a·b}$$