If a is a rational number, multiplied by itself m times, it is written as am. Here, a is called the base and m is called the exponent.
Exponential Notion: The notation for writing the product of a number by itself several times. For example,
a × a × a × a = a4
Base and Exponent: an = a × a × a (n times). Here, a = base, n = exponent.
Prime factorisation: Any natural number other then1, can be expressed as a product of power of prime numbers.
Laws of Exponents
Positive Integers as Exponents
Law 1: If a is any non-zero rational number and m and n are two positive integers, then
$$ a^m \times a^n = a^{m+n} $$
Law 2: If a is any non-zero rational number and m and n are positive integers (m > n), then
$$ \frac{a^m}{a^n} = a^{m-n} $$
Law 3: When n > m
$$ \frac{a^m}{a^n} = \frac{1}{a^{m-n}} $$
Law 4: If a is any non-zero rational number and m and n are two positive integers, then
$$ (a^{m})^{n} = a^{mn} $$
Law 5 (Zero Exponent): If a is any rational number other than zero
$$ a^{0} = 1 $$
Negative Integers as Exponents
$$ a^{-n} = \frac{1}{a^{n}} $$
Radicals or Surds
$ \sqrt[n]{x} $ is a surd if and only if it is an irrational number and it is a root of the positive rational number. The index n is called the order of the surd and x is called the radicand.
Pure and mixed surd: A surd with rational factor 1 only, other factor being irrational is called pure surd. A surd having rational factor other then 1 along with the irrational factor is called a mixed surd.
Similar or like surds: Two surds are said to be similar if they have same irrational factor.
Comparison of Surds: Change the given surds to surds of the same order, then compare their radicands along with co-efficient.
Laws of Surds
$$ \sqrt[n]{x} \sqrt[n]{y} = \sqrt[n]{xy} $$
$$ \sqrt[n]{x^m} = x^\frac{m}{n} $$