CAT 2025 Lesson : Surds & Indices - Indices Rules

2. Indices

Numbers or variables written in the form $a^b$ means a needs to be multiplied b times.
$a^n = a \times a \times a \times ..._{n\ times}$
$\therefore 3^4 = 3 \times 3 \times 3 \times 3 = 81$

In $a^b$, $a$ is called the base and $b$ is called the power, exponent or index.
(Indices is the plural for index)

The following are the rules for indices

S.No.	Rule	Example
1	$a^0 = 1$	$3^0 = 24^0 = (-137)^{0} = 1$
2	$a^1 = a$	$4^1 = 4, 75^1 = 75$
3	$1^n = 1$	$1^5 = 1^{795} = 1^{-15} = 1$
4	$a^m \times a^n = a^{m + n}$	$3^5 \times 3^7 = 3^{5 + 7} = 3^{12}$
5	$\dfrac{a^m}{a^n} = a^{m-n}$	$\dfrac{3^5}{3^7} = 3^{5 - 7} = 3^{-2}$
6	$a^{-n} = \dfrac{1}{a^n}$	$3^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9}$
7	$({a^m})^n = a^{m \times n}$	$({5^3})^2 = 5^{3 \times 2} = 5^6$
8	$\large{{a^m}^{n}} = \large{a}^ {({m}^{n})}$	$\large{{5^3}^2} = 5^9$
9	$a^n \times b^n = ({ab})^n$	$13^8 \times 2^8 = 26^8$
10	$\dfrac{{a}^n}{{b}^n} = \left( \dfrac {a}{b} \right)^n$	$\dfrac{{13}^8}{{2}^8} = \left( \dfrac {13}{2} \right)^8$
11	${a}^\frac{m}{n} = \sqrt[n]{a^m}$	${5}^\frac{4}{3} = \sqrt[3]{5^4}$

To extend rule $\bm{8}$, if the powers are raised one over the other then, we start with applying the top most power and then move downwards.

${{3^2}^3}^2 = {3^2}^9 = 3^{512}$

If brackets/parentheses would have been used, then the answer would have been different. We would apply rule $\bm{7}$.

$((3^2)^3)^2 = 3^{2 \times 3 \times 2} = 3^{12}$

Example 6

Simplify $8^\frac{2}{3} \times 5^3 - ( 2^3 )^2 \div 3$

Solution

$= ( 8^2 )^\frac{1}{3} \times 125 - \dfrac{2^6}{3}$

$= ( 64 )^\frac{1}{3} \times 125 - \dfrac{64}{3} = ( 4^3 )^{\frac{1}{3}} \times 125 - \dfrac{64}{3}$

$= 4 \times 125 - \dfrac{64}{3}$ $= 500 - \dfrac{64}{3}$ $= \dfrac{1436}{3}$

Answer: $\dfrac{1436}{3}$