Surds and Indices Problems and Solutions Formulas, Tips and Tricks
1. Laws of Indices
\begin{aligned}
i).& a^m \times a^n = a^{m+n} \\
ii).& \frac{a^m}{a^n} = a^{m-n} \\
iii).& (a^m)^n = a^{mn} \\
iv).& (ab)^n = a^n b^n \\
v).& \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \\
vi).& a^0 = 1 \\
\end{aligned}2. Surds formula
Let a be a rational number and n be a positive integer such that \begin{aligned} a^{\frac{1}{n}} = \sqrt[n]{a}
\end{aligned} is irrational.
then, \begin{aligned} \sqrt[n]{a} \end{aligned} is called a surd of order n.3. Laws of Surds
\begin{aligned}
i.) & \sqrt[n]{a} = a^{\frac{1}{n}} \\
ii.) & \sqrt[n]{ab} = \sqrt[n]{a} * \sqrt[n]{b} \\
iii.) & \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{ \sqrt[n]{b}} \\
iv.)& (\sqrt[n]{a})^n = a \\
v.)& (\sqrt[m]{\sqrt[n]{a}}) = \sqrt[mn]{a} \\
vi.)& (\sqrt[n]{a})^m = \sqrt[n]{a^m}
\end{aligned}