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, $3^4=$ $3 \times 3 \times 3 \times 3$
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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 $a=b$
