Indices & Surds: Definition of Indices

by Wonder Boy · Published September 19, 2021 · Updated October 7, 2021

An index (plural: indices) is the power or exponent of a number.
For Example: $p^5$ has an index of 5.

Here, by the term $p^5$ we basically mean p x p x p x p x p i.e product of p 5 times.

Similarly, by the term $a^n$ we basically mean a x a x a … n times. i.e. product of a n times.

Here, a is called the base and n is called the power or index of the number $a^n$ .

In the term $a^{\frac{1}{n}}$ , a is the base and $\frac{1}{n}$ is the power or index of the number $a^{\frac{1}{n}}$ . It may also be called as nth root of a, denoted by $\sqrt[n]{a}$ .

The manipulation of indices can be a powerful tool to simplify an expression. There are various laws of indices which govern the whole operation.

1. Multiplication Rule:
$x^m$ x $x^n$ = $x^{m + n}$

For Example:
$3^3$ x $3^4$ = $3^{3 + 4}$ = $3^7$

Here, L.H.S = $3^3$ x $3^4$ = 27 x 81 = 2187
R.H.S = $3^7$ = 2187
L.H.S = R.H.S = 2187

2. Multiplication Rule:
$x^m$ x $x^n$ x $x^p$ x …… = $x^{m + n + p+....}$

For Example:
$3^2$ x $3^3$ x $3^4$ = $3^{2+3+4}$ = $3^9$

Here, L.H.S. = $3^2$ x $3^3$ x $3^4$ = 9 x 27 x 81 = 19683
R.H.S. = $3^9$ = 19683
L.H.S = R.H.S = 19683

3. Division Rule:
$\frac{x^m}{x^n}$ = $x^{m-n}$

For Example:
$\frac{3^3}{3^2}$ = $3^{3-2}$ = $3^1$ = 3

Here, L.H.S. = $\frac{3^3}{3^2}$ = $\frac{27}{9}$ =3
R.H.S. = $3^{3-2}$ = $3^1$ = 3
L.H.S. = R.H.S. = 3

4. Power Rule:
$(x^m)^n$ = $x^{mn}$

For Example:
$(3^2)^3$ = $3^{2*3}$

Here, L.H.S. = $(3^2)^3$ = $9^3$ = 729
R.H.S. = $3^{2*3}$ = $3^6$ = 729
L.H.S. = R.H.S. = 729

5. Power Rule:
$x^{\frac{p}{q}}$ = $\sqrt[q]{x^p}$

For Example:
$3^{\frac{2}{5}}$ = $\sqrt[5]{3^2}$ = $\sqrt[5]{9}$

6. Power Rule:
$x^{\frac{1}{n}}$ = $\sqrt[n]{x}$

For Example:
$3^{\frac{1}{3}}$ = $\sqrt[3]{3}$

7. $x^{-n}$ = $\frac{1}{x^n}$

For Example:
$5^{-3}$ = $\frac{1}{5^3}$

8. $x^0$ = 1 (where x $\neq$ 0)

For Example:
$5^0$ = 1

9. $(x * y)^m$ = $x^m$ x $y^m$

For Example:
$(3 * 5)^2$ = $3^2$ x $5^2$ = 9 x 25 = 225

Here, L.H.S. = $(3 * 5)^2$ = $15^2$ = 225
R.H.S. = $3^2$ x $5^2$ = 9 x 25 = 225.

10. $(\frac{x}{y})^m$ = $\frac{x^m}{y^m}}$

For Example:
$(\frac{3}{4})^2$ = $\frac{3^2}{4^2}}$ = $\frac{9}{16}}$

Here, L.H.S. = $(\frac{3}{4})^2$ = $(0.75)^2$ = 0.5625
R.H.S. = $\frac{3^2}{4^2}}$ = $\frac{9}{16}}$ = 0.5625
L.H.S. = R.H.S. = 0.5625

11. If $x^m$ = $x^n$ and x $\neq$ -1, 0 , 1 then m = n.

For Example:
$3^m$ = $3^n$
Here, 3 $\neq$ -1, 0 , 1
So, m = n