2 Laws of indices

There is a set of rules which enable us to manipulate expressions involving indices. These rules are known as the laws of indices , and they occur so commonly that it is worthwhile to memorise them.

Key Point 5
Laws of Indices

The laws of indices state:

First law :  a m × a n = a m + n add indices when multiplying numbers with the same base

Second law : a m a n = a m − n  subtract indices when dividing numbers with the same base

Third law : ( a m ) n = a m n   multiply indices together when raising a number to a power

Example 16

Simplify

  1. a 5 × a 4 ,
  2. 2 x 5 ( x 3 ) .
Solution

In each case we are required to multiply expressions involving indices. The bases are the same and we use the first law of indices.

  1. The indices must be added, thus a 5 × a 4 = a 5 + 4 = a 9 .
  2. Because of the associativity of multiplication we can write

    2 x 5 ( x 3 ) = 2 ( x 5 x 3 ) = 2 x 5 + 3 = 2 x 8

The first law of indices (Key Point 5) extends in an obvious way when more terms are involved:

Example 17

Simplify b 5 × b 4 × b 7 .

Solution

The indices are added. Thus b 5 × b 4 × b 7 = b 5 + 4 + 7 = b 16 .

Task!

Simplify y 4 y 2 y 3 .

All quantities have the same base. To multiply the quantities together, the indices are added: y 9

Example 18

Simplify

  1. 8 4 8 2 ,
  2. x 18 ÷ x 7 .
Solution

In each case we are required to divide expressions involving indices. The bases are the same and we use the second law of indices (Key Point 5).

  1. The indices must be subtracted, thus 8 4 8 2 = 8 4 − 2 = 8 2 = 64 .
  2. Again the indices are subtracted, and so x 18 ÷ x 7 = x 18 − 7 = x 11 .
Task!

Simplify 5 9 5 7 .

The bases are the same, and the division is carried out by subtracting the indices: 5 9 − 7 = 5 2 = 25

Task!

Simplify y 5 y 2

y 5 − 2 = y 3

Example 19

Simplify

  1. ( 8 2 ) 3 ,
  2. ( z 3 ) 4 .
Solution

We use the third law of indices (Key Point 5).

  1. ( 8 2 ) 3 = 8 2 × 3 = 8 6
  2. ( z 3 ) 4 = z 3 × 4 = z 12 .
Task!

Simplify ( x 2 ) 5 .

x 2 × 5 = x 10

Task!

Simplify ( e x ) y

Again, using the third law of indices, the two powers are multiplied: e x × y = e x y

Two important results which can be derived from the laws of indices state:

Key Point 6
Any non-zero number raised to the power 0 has the value 1, that is a 0 = 1
Any number raised to power 1 is itself, that is a 1 = a

A generalisation of the third law of indices states:

Key Point 7
( a m b n ) k = a m k b n k
Example 20

Remove the brackets from

  1. ( 3 x ) 2 ,
  2. ( x 3 y 7 ) 4 .
Solution
  1. Noting that 3 = 3 1 and x = x 1 then ( 3 x ) 2 = ( 3 1 x 1 ) 2 = 3 2 x 2 = 9 x 2



    or, alternatively ( 3 x ) 2 = ( 3 x ) × ( 3 x ) = 9 x 2



  2. ( x 3 y 7 ) 4 = x 3 × 4 y 7 × 4 = x 12 y 28
Exercises
  1. Show that ( − x y ) 2 is equivalent to x 2 y 2 whereas ( − x y ) 3 is equivalent to − x 3 y 3 .
  2. Write each of the following expressions with a single index:
    1. 6 7 6 9 ,
    2. 6 7 6 19 ,
    3. ( x 4 ) 3
  3. Remove the brackets from
    1. ( 8 a ) 2 ,
    2. ( 7 a b ) 3 ,
    3. 7 ( a b ) 3 ,
    4. ( 6 x y ) 4 ,
  4. Simplify
    1. 15 x 2 ( x 3 ) ,
    2. 3 x 2 ( 5 x ) ,
    3. 18 x − 1 ( 3 x 4 ) .
  5. Simplify
    1. 5 x ( x 3 ) ,
    2. 4 x 2 ( x 3 ) ,
    3. 3 x 7 ( x 4 ) ,
    4. 2 x 8 ( x 11 ) ,
    5. 5 x 2 ( 3 x 9 )
    1. 6 16 ,
    2. 6 − 12 ,
    3. x 12
    1. 64 a 2 ,
    2. 343 a 3 b 3 ,
    3. 7 a 3 b 3 ,
    4. 1296 x 4 y 4
    1. 15 x 5 ,
    2. 15 x 3 ,
    3. 54 x 3
    1. 5 x 4 ,
    2. 4 x 5 ,
    3. 3 x 11 ,
    4. 2 x 19 ,
    5. 15 x 11