Roots and Power

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


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 \(\sqrt[3]{8}=2\) 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

  • \(a^{n}·a^{m} = a^{n + m}\)

  • \(\frac{a^{n}}{a^{m}} = a^{n-m}\)

  • \(a^{n}·b^{n}=(ab)^{n}\)

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

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

  • \((a^n)^m=a^{nm}\)

  • \(a^0=1\)

  • \(\sqrt[n]{1}=1\)

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

  • \(\displaystyle\frac{a}{\sqrt{a}}= \sqrt{a}\)

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

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