Index: Definition, Laws of Indices, and Solved Examples

The index of a number is also known as the power or exponent. It actually tells us how many times we have to multiply the number by itself. For example, we consider $3^4.$ It means we have to multiply 3 by itself four times. In other words, $3^4=$ $3 \times 3 \times 3 \times 3$

Definition of Index

The power/exponent raised to a number is called the index of that number.

Mathematically, we understand the index of a number as follows. For a real number $a$ and for a positive integer $n$, we have \[a^n=a \times a \times \cdots \times a \,(m \text{ times})\] Here the number $n$ is called the index of $a$ and the number $a$ is called the base. In the above example $3^4=$ $3 \times 3 \times 3 \times 3,$ the number $4$ is the index of $3$, and the number $3$ is the base.

Note that the plural form of index is indices. ♣

 

Laws of Indices

We now discuss the laws of indices (or the rules of indices). This will help us to solve the problems of indices.

• Zero index rule: $a^0=1$

Thus, if the index of any non-zero number is $0,$ then the value will be $1.$ For example, $1^0=1, 7^0=1.$ But, remember that $0^0 \neq 1.$ Note $0^0$ is meaningless.

 

• Negative index rule:$a^{-n}=\frac{1}{a^n}$

Thus, if the index of any number is negative, then the value will be the reciprocal of the positive index raised to that number itself. For example, $7^{-1}=\frac{1}{7^1}=\frac{1}{7}$

 

• Quotient rule of indices: 

(i) $a^m \div a^n$ $=\frac{a^m}{a^n}$ $=a^{m-n},$ $\quad$  (ii) $(\frac{a}{b})^n$ $=\frac{a^n}{b^n}$

Examples:

(i) $2^4 \div 2^2$ $=\frac{2^4}{2^2}$ $=2^{4-2}, \quad$ (ii) $(\frac{4}{2})^2$ $=\frac{4^2}{2^2}$

 

• Product rule of indices: 

(i) $a^m . a^n$ $=a^{m+n},$ $\quad$ (ii) $(a^m)^n$ $=a^{mn},$ $\quad$ (iii) $(ab)^n$ $=a^n . b^n$

Examples:

(i) $2^2 . 2^3=2^{2+3}, \quad$ (ii) $(2^3)^2$ $=2^{3 . 2}, \quad$ (iii) $(2 . 3)^2$ $=2^2 . 3^2$

 

• Fraction rule of indices: $a^{\frac{m}{n}}=\sqrt[n]{a^m}$

So if the index is a fraction, then the value can be expressed as a radical form. For example, $7^{\frac{1}{3}}$ $=\sqrt[3]{7^1}$ $=\sqrt[3]{7}$ (cube root of $7$)

 

Some Remarks on Indices

(R1) Simple proof of $a^0=1$

Proof: $a^0=$ $a^{m-m}$ $[\because 0=m-m]$

$=\frac{a^m}{a^m}$ $[\because a^{x-y}=\frac{a^x}{a^y}]$

$=1$

(R2) Meaning of $a^n$ when $n$ is a negative integer: write $n=-m,$ where $m$ is a positive integer. So $a^n=$ $a^{-m}$ $=\frac{1}{a^m}$ (negative index rule)

(R3) Meaning of $a^n$ when $n$ is a fraction: we write $n=\frac{p}{q},$ where $p$ is an integer (positive or negative) and $q$ is a positive integer. So $a^n=$ $a^{\frac{p}{q}}$ $=(a^p)^{\frac{1}{q}}$ $=\sqrt[q]{a^p}$ (fraction rule of indices)

 

Before providing solved examples, we now summarize the fundamental laws of indices.

$a^0=1$

$a^{-n}=\frac{1}{a^n}$

$a^m . a^n=a^{m+n}$

$a^m \div a^n=a^{m-n}$

$(a^m)^n=a^{m \times n}$

$(ab)^n=a^n . b^n$

$(\frac{a}{b})^n=\frac{a^n}{b^n}$

$a^{\frac{1}{n}}=\sqrt[n]{a}$

$a^{\frac{m}{n}}=\sqrt[n]{a^m}$

 

Solved Examples of Indices

Ex 1: Simplify $8^{2/3}$

Solution. 

Note that $8=2 \times 2 \times 2=2^3$

∴ $8^{2/3}=$ $(2^3)^{2/3}$ $=2^{3 \times 2/3}$ $=2^2$ $=4$

 

Ex 2: Calculate $\{(2^{-7})^{\frac{3}{7}}\}^{\frac{1}{3}}$

Solution. 

$\{(2^{-7})^{\frac{3}{7}}\}^{\frac{1}{3}}$

$=\{2^{-7 \times \frac{3}{7}}\}^{\frac{1}{3}}$

$=(2^{-3})^\frac{1}{3}$

$=2^{-3 \times \frac{1}{3}}$

$=2^{-1}$

$=\frac{1}{2}$

 

Ex 3: If $x^a=x^b,$ then $a=b?$

Solution.

Case 1: $x \neq 0$

Now $x^a=x^b$ ⇒ $\frac{x^a}{x^b}=1$ ⇒ $x^{a-b}=1$

So $a-b=0$ as $x \neq 0$

⇒ $a=b$

Case 2: $x=0$

Note that $0^2=0^5=0,$ but $2 \neq 5$

Conclusion: $x^a=x^b$ does not always imply that $a=b$