**5.1 Indices**

**5.1.1 Indices**

**1.**A number expressed in the form

*a*is known as an index notation.

^{n}**2.**

*a*is read as ‘

^{n}*a*to the power of

*n*’ where

*a*is the

**base**and

*n*is the

**index**.

*Example:***If**

3.

3.

**is a real number and**

*a***is a positive integer, then**

*n***4.**The value of a real number in index notation can be found by

**repeated multiplication**.

*Example:* 6

^{4 }= 6 × 6 × 6 × 6 = 1296

**A number can be expressed in index notation by dividing the number repeatedly by the base.**

5.

5.

*Example:*243 = 3 × 3 × 3 × 3 × 3

**5.1.2 Multiplication of Numbers in Index Notation**

The multiplication of numbers or algebraic terms with the

**same base**can be done by using the**Law of Indices**.

*a*^{m }

^{ }**×**

*a*^{n}^{ }=*a*^{m}^{ + n}

*Example:*3

^{3 }× 3^{8 }= 3^{3+8} = 3

^{11 }**5.1.3 Division of Numbers in Index Notation**

**1.**The Law of indices for division is:

*a*^{m }

^{ }**÷**

*a*^{n}^{ }=*a*^{m}^{ - n}

*Example:*4

^{12 }÷ 4^{12 }= 4^{12-12 }= 4

^{0}= 1

**2.**

*a*^{0}^{ }= 1**5.1.4 Raise Numbers and Algebraic Terms in Index Notation to a Power**

To raise a number in index notation to a power, multiply the two indices while keeping the base unchanged.

**(**

*a*)^{m }^{ n }^{ }= (*a*)^{n }^{ m }^{ }=*a*^{m}^{n}

*Example:*(4

= 4^{3})^{7}= 4^{3}×^{7}^{21 }

**5.1.5 Negative Indices**

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

Example:Example:

**5.1.6 Fractional Indices**

${a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}}\text{or}{\left(\sqrt[n]{a}\right)}^{m}$

*Example:*