Indices (singular Index) is the exponent of any base. The index or exponent is the number that shows how many the base is raised to the power of any number or how many time(s) the base is multiplied by itself. There are some laws of indices that come really handy when solving algebraic expressions.
Indices
In an algebraic term, we can have base, coefficient, and power. The power of a base in an expression is called its index. The plural form of index is indices.
In algebra, Indices is the representation of the power of the base of an algebraic term.
For example:
We have, $\rm a \times a$. We write $\rm a \times a$ as $\rm a^{2}$. This '2' written as the exponent after a represents the base 'a' and is raised to the power of '2'.
Laws of Indices
There are some laws of indices that come really handy when solving expressions. These laws are the proven rules that can be applied to solve any algebraic expressions.
Here are some of the laws of indices according to different conditions:
Multiplication Law
$\rm a^{m} \cdot a^{n} = a^{m + n}$
Division Law
$\rm a^{m} \div a^{n} = a^{m-n}$
Product Law
$\rm \left ( a^{m} \right )^{n} = a^{mn}$
$\rm \left ( ab \right ) ^{m} = a^{m} \cdot b^{m}$
$\rm \left ( \frac{a}{b} \right )^{m} = \frac{a^{m}}{b^{m}}$
Law of Negative Exponents
$\rm a^{-m} = \frac{1}{a^{m}}$
$\rm a^{m} = \frac{1}{ a^{-m}}$
Law of Zero Power
$\rm a^{0} = 1$
$\rm 100^{0} = 1$
Root Law
$\rm \sqrt[n]{ a^{m} } = a^{ \frac{m}{n} }$