INDICES AND ITS LAWS

Quote:
Minds are like parachutes. They only functions when they are open. SIR JAMES DEWAR (scientist 1877 - 1925)

INDICES

A number is said to be in an index form when it is written in the form aⁿ where ‘a’ is called the base and ‘n’ which may be positive or negative is called the power, exponent or index. For example, a shot form of writing 2x2x2x2x2x2x2 is 2⁷, 2 is the base and 7 is the power, exponent or power.

Basic Laws of Indices  

•   Multiplication Law: If two or more terms of the same base with equal or different powers are multiplied, the result is the common base raised to the sum of the powers. In algebraic form we have:

            aⁿ   x  a^m  =  a^m+n

Similarly,        aⁿ  x   bⁿ =  (ab)ⁿ

Example: X²Y⁴Z³  *  X⁴YZ²     = X²⁺⁴ Y⁴⁺¹ Z³⁺²    =  X⁶ Y⁵ Z^{5   
note: the above is done by bring like terms together, then adding up their index or power)}

               2² . 2⁴  = 2²⁺⁴ = 2⁶

               5² . 2² = (5.2)² = 10² = 100

NOTE: (.) MEANS MULTIPLICATION   

•  Division Law: If two terms of the same base is divided, the result is the common base raised to the difference in their powers; Note that the power of the denominator is taken from that of the numerator. Algebraically, we have

            a^m   ÷   aⁿ =   a^m = a^{m – n}

                               aⁿ

      Similarly,    a^m  ÷  b^m   =    a ^m

                                                  b

Example:  4⁷÷4³ = 4^{7- 3}   = 4⁴ = 256

                   
X⁸ Y⁴   ÷  X⁵ Y²  = X^{8- 5}   Y^{4- 2}  = X³  Y²

                  15³ ÷ 5³ =  15 ³ = 3³ = 27

                                        5

• Power / Function into function Law: Algebraically, this is expressed as:

                                                    (a^m)ⁿ = a^mn

Example: (2⁵)² = 2¹⁰

This law is an extension of multiplication law. 

•   Fractional Index Law: a^m/n  = (ⁿ√a)^m

Example:        64^2/3 = (³√64)⁴ = 4² = 16

                  Z^3/7 simply means (³√Z)

•    Reciprocal/Negative index Law:

a^-n = 1

        aⁿ     

Example:  3^-5 = 1

                           3⁵ 

•  Zero Law:  Z⁰ = 1 for all values of Z is not equal to zero

Example: (XYZ)⁰ = 1  

                  (-25)⁰ = 1

NOTE: THESE LAWS CAN BE COMBINED TO SOLVE PROBLEMS THAT INVOLVE THE USE OF INDICES.

IN THE NEXT POST, WE WILL TREAT DIFFERENT QUESTIONS REGARDING THIS LAWS OF  INDICES.

Thanks 
Emmanuel Bassey 
FOR: qedmaths
Edited by
Imoh Friday
  you too can have your topic posted here , all you have to do is to send your topic content to qedmathematics@gmail.com