Law of Indices

Indices is a number with the power. For example: a^m; a is called the base and m is the power. These laws only apply to expression with the same base.

Index help to write a product of numbers very compactly. Index help to show how many times to use the number in a multiplication. It is shown in the top right of the number in small number.

In this example: 4³ = 4x4x4 = 64

Rule1: a° = 1

Any number, except 0, whose index is 0 is always equal to 1.

An example:

2° = 1

Rule 2: a^-m = \(\frac{1}{a^m}\)

An example:

2^-3 = \(\frac{1}{2^3}\) ( using a^-m = \(\frac{1}{a^m}\))

Rule 3: a^mx aⁿ = a^m+n

In case of multiplication of same base, copy the base and add the indices.

An example:

3² x 3⁴ = 3²⁺⁴ (using a^m x aⁿ = a ^m+n)

= 3⁶

= 3 x 3 x 3 x 3 x 3 x 3

= 729

Rule 4: a^m ÷ aⁿ = a^m-n

In case of division of same base, copy the base and subtract the indices.

An example:

w¹⁰ ÷ w⁶= w^10-6 = w⁴

Rule 5: ( a^m)ⁿ = a^mn

To raise an expression to the n^th index, Copy the base and multiply the indices.

An example:

( x²)⁴ = x^2x4 = x⁸

Rule 6: a^{\(\frac{m}{n}\)} = (\(\sqrt[n]{a}\))^m

An example:

125^{\(\frac{2}{3}\)} = (\(\sqrt[3]{125}\))² = (5)² = 25