By : Normaisyarah bt Zainudin

Nur Syahmi bt Nazri

( PPISMP Mathematics’10 )

INDICES :

We know that:     5 × 5 = 25

The product 5 × 5 can be written as 52.

5 × 5 is known as the expanded form (or factor form) of 25 and 52 is known as the index form of 25.

Generally when a number is multiplied by itself any number of times, the expression is simplified by using the index notation.

Note: 2 is called the base. 3 is called the index or power (or exponent) because it indicates the power to

which the base, 2, is raised. 8 is the basic numeral (or number). 23 is read as '2 to the power 3' or simply '2 cubed'.

That is:

Example 1

Write 43 as a number.

Solution:

Note the following: 64 = 43

3 is the power (or index or exponent) 4 is the base number 64 is a basic numeral or number 43 is the index form (or power form) of 64 4 × 4 × 4 is the expanded form of 64 For 64 = 4 × 4 × 4 = 43, the base number 4 appears three times as a factor of the

basic numeral (or number) 64 43 is read as '4 to the power 3' or simply '4 cubed'

Example 2

Write each of the following expanded forms in index form:

Solution:

Example 3

Write each of the following in expanded form:

Solution:

Example 4

Find the value of the following:

Solution:

Example 5

Write 16 in index form using base 2.

Solution:

LAW OF INDICES :

Large or small numbers are better expressed in terms of indices. A given number is written as a base raised to the index, that is (base) index.  Bases and indices can be any real number. In the last chapter we have seen what are squares and cubes of numbers; for squares the index is 2, for cubes the index is 3. For square roots the index is (1/2) and for cube roots the index is (1/3). The base has to be a suitable number that together with the indices gives the correct number.   If the index is n, then the resultant number is obtained by multiplying the base n times.

Laws of indices

This formula tells us that when multiplying powers with the same base, add the indices.This is the first index law and is known as the Index Law for Multiplication.

We will state a few facts about indices and try to see their validity using all sorts of numbers. The laws are valid for all real numbers, but for the present syllabus, it is sufficient to consider only rational numbers.   

1. am * an = a (m+n).      

2. a (-m) = 1 / am.

3. am / an = a (m-n) , a 0.

4. (am) n    =    a (m * n).

5. (a * b) m = am * bm.  

6. (a / b) m = am / bm , b 0.

7. a0 = 1.

1. To prove that am * an = a (m+n).  

Example 1 :  Let us consider the base a = 2, let the value of indices m and n be m=4, n=5.  

LHS : am * an = 24 * 25 = (2 * 2 * 2 * 2) * (2 * 2 * 2 * 2 * 2) = 16 * 32 = 512.  

RHS : a (m+n)  = 29 = 512.

Since LHS = RHS, hence it is proved that am * an = a (m+n).

Example 2 : Let us consider the  base a = 10, let the value of indices m and n be m=2, n=3.

LHS : am * an = 102 * 103 = (10 * 10) * (10 * 10 * 10) = 100 * 1000 = 100000.

RHS : a (m+n)  = 105 = 100000.

Since LHS = RHS, hence it is proved that am * an = a (m+n). 

2. The second law of indices

This formula tells us that when dividing powers with the same base, the index in the denominator is subtracted from the index in the numerator.

This is the second index law and is known as the Index Law for Division.

To prove that a (-m) = 1 / am.

Multiply both the LHS and the RHS by am 

a (-m)  * am  =  (1 / am) *  am.

From first law of indices proved above, LHS becomes a (m-m) = a0.

In the RHS, the numerator and the denominator cancel each other out to give a resultant of 1.

Thus a (m-m) = 1, which means that a (-m) * am = 1. This gives a (-m) = 1/ am.       

Example:

Solution:

Note:

Simplify the numerical coefficients first, and then apply the index law.

3. From the above two proofs, the third law of indices

         am / an = a (m-n) ,  a 0.

can be easily deduced.

The condition a 0 is important, because if a = 0, the expression will become infinite.

If m > n, the index of the expression on the RHS, that is (m-n), will be a positive number.

If  m < n, the index of the expression on the RHS, the is (m-n), will be a negative number. In this case the second law of indices will have to be applied to obtain the value.  

4. The 4 th law of indices

To prove that (am) n = a (m * n)

Example 1 : Let us consider the  base a = 7, let the value of indices m and n be m=2, n=5.

LHS : (am) n = (72) 5 = ( 7 * 7 ) 5  = 495  = 49 * 49 * 49 * 49 * 49 = 282475249 .

RHS : a (m * n) = 710  =  7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7  =  282475249 .

Thus, LHS = RHS, hence it is proved that (am) n = a (m * n) .

Example 2 : Let us consider the  base a = 1/2, let the value of indices m and n be m=1, n=2.

LHS : (am) n    = ( (1/2)1 )2 =   (1/2)2  = ‡ * ‡  = º .

RHS : a (m * n) =  (1/2)2  = ‡ * ‡  = º .

Thus LHS = RHS, hence it is proved that (am) n = a (m * n).

5. The 5 th law of indices

To prove that  (a * b) m = am * bm.   

Example 1 : Let us consider the base a = 2 and b = 3, let the value of index m be m = 2 .

LHS : (a * b) m = ( 2 * 3 )2 = 62 = 6 * 6 = 36 .

RHS : am * bm   = 22 * 32 = 4 * 9 = 36 .

Thus LHS = RHS, hence it is proved that (a * b) m = am * bm. 

Example 2 : Let us consider the base a = 4 and b = 9, let the value of index m be m = ‡ .

LHS : (a * b) m = ( 4 * 9 )1/2 = 361/2   = 6 .

RHS : am * bm   =  41/2   * 91/2   = 2 * 3 = 6.

Thus LHS = RHS, hence it is proved that (a * b) m = am * bm.    

6. From the 1st , 2nd and the 5th laws of indices the 6 th law of indices can also be proved. Thus,

     (a/b)m = am / bm ,   b 0.    

The condition b 0 is necessary, otherwise the expression will become infinite with 0 in the denominator.

7. The 7 th law of indices a0 = 1 is already proved above in the 2nd  law of indices.

 

Summary The indices are a very easy and convenient method for expressing very large or very small numbers. In case of small numbers, the index is negative and in case of large numbers the index is positive. All the laws of indices are useful in many calculations. These laws are valid for all real numbers, whether integers or fractions.