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Index Notation & Index Laws

We saw in index notation in Pre-Algebra on how to represent numbers using basic index notation. We’ll now look at how we can use them to execute various mathematical calculations, such as multiplication and division.

When you multiply two pronumerals together, or multiply a pronumeral to itself by a certain number of times, we write the expression in an index form for the repeated pronumeral. This representation is called index notation.

We know y x y is y2. The 2 in y2 is called the power or index, and y is the base.

Similarly  p x p x p x p  =  p4.

Here p is the base, and 4 is the index (or power) of p. Expressing numbers with a base and an index is called index notation.

We’ll now look at a number of index laws

 

Index law for multiplication

This law states that when you have two index numbers to multiply

  1. multiply any co-efficients first, and then
  2. keep the same base, and add the indices

In other words, xn x xm  =  xn+m, and  axn x bxm  =  abxn+m

Example 1:  23 x 24

=  (2 x 2 x 2) x (2 x 2 x 2 x 2)

=  27

Example 2:  a3 x a5

=  (a x a x a) x (a x a x a x a x a)

=  a8

Example 3:  2a2 x 3a3

=  (2 x a x a)  x  (3 x a x a x a)

=  (2 x 3)  x  (a x a x a x a x a)

=  6a5

 

Index law for division

This is the opposite of index law for multiplication. When you divide two index numbers, you should obey the following rules:

  1. divide any co-efficients first, and then
  2. keep the same base, and subtract the indices

In other words, xn ÷ xm  =  xn-m, and  {ax^n}/{bx^m}  =  a/bxn-m

Example 1:  26 ÷ 22

=  {2~*~2~*~2~*~2~*~2~*~2}/{2~*~2}

=  2 x 2 x 2 x 2

=  24

Example 2:  y7 ÷ y3

=  {y~*~y~*~y~*~y~*~y~*~y~*~y}/{y~*~y~*~y}

=  y~*y~*~y~*~y

=  y4

Example 3:  12a5 ÷ 4a4

=  12/4 x {a~*~a~*~a~*~a~*~a}/{a~*~a~*~a~*~a}

=  3a

 

Index law for powers

To simplify an expression with further power of indices, you should obey the following rules:

  1. raise any co-efficient to the power outside the grouping symbols, and
  2. keep the same base, and multiply the indices

In other words, (x^m)^n  =  xmn, and  (ax^m)^n  =  an.xmn

Example 1:  (2^3)^2

=  23 x 23

=  23+3  (using the index law of multiplication)

=  26

=  23×2  =  26

Example 2:  (a^2)^3

= a2 x a2 x a2

= a2+2+2  (using the index law of multiplication)

= a6

= a2×3  =  a6

Example 3:  (2a^2)^3

=  2a2 x 2a2 x 2a2

=  (2 x 2 x 2)  x  (a2 x a2 x a2)   (using the index law of multiplication)

=  8 x a6

=  8a6

 

Index law for power 0

This is a special type of an index law for powers – how to calculate the expression involving an index (or power) of 0?

Using the index laws for division, we can see that {x^m}/{x^m} = 1, since

xm-m  =  x0  =  1

Hence we get x0  =  1, i.e. any number or expression to the power of zero is 1.

Note: 1. Another name for index is power or exponent.

2. The scientific calculator button for power is

 

Press base ‘number’ x, then the power button, then enter the ‘index’ y.

 

Summary of Index Laws & Index Notation

Index law for multiplication

xn  x  xm  =  xn+m, andaxn  x  bxm  =  abxn+m

Index law for division

xn  ÷  xm  =  xn-m, and{ax^n}/{bx^m}  =  a/bxn-m

Index law for powers

(x^m)^n  =  xmn, and(ax^m)^n  =  an.xmn

Index law for power 0

x0  =  1

 

In addition, there are two other laws of index – law for negative indices, and law for fractional indices