Learn Mathematics: Indices

Indices are small numbers written above a larger numbers to indicate or show how many times the larger number must be multiplied by itself. For example, 

4² = 4 x 4 = 16 = 4 x 4. 

The superscript (i.e. number  2  above 4) is known as index while 4 is the base.

Laws of indices: 

There are laws of indices one must strictly observe before proceeding. These are:

(1) Multiplication Law: When numbers with equal bases are being multiplied, their powers (indices) are added.

             Example:

                             3² x 3⁵ = (3x3)x(3x3x3x3x3)

                                             = 3² + ⁵

                                                                   ⁼  3⁷

But if two groups of factors ( i.e. different numbers) are multiplied, the different factors (multiples of a number) are collected as one group to give the product.

Example:    

                                =    2⁵ x 3^{2 x} 2³ _x 3⁴ 3⁵

                                 =   2⁵x2³x3²X3⁴

                                 =   2^{5 + 3} x 3^{2 + 4}

       

                                  =   2⁸ x 3⁶

(2) Division Law: When dividing numbers with equal bases, the powers of the divisor is subtracted from that of the dividend. Example: 2⁹ -:- 2⁶

                                                    =  2^9-6

                                               =  2³

         OR

                                 2x2x2x2x2x2x2x2x2

                                ---------------------------  

                                     2x2x2x2x2x2

                               = 2³      

                        

(3) Power Law: If a number which is raised to a certain power is again raised to another power, it is said to be in powers. The two powers are thus multiplied together for final result. Example: (2³)²                                     

                               =  2³x2³

                               =  2x2x2 X2x2x2

                            =         2⁶                                  

       OR                   = ( 2³)²

                              =  2^3x2       

                               =  2⁶.

 

Problem solving:

 solve the following problems.

(1.)           2⁶x5⁴ x2²5²

(2.)           6^10-:-6⁵

Solution

                          2⁶x5⁴x2²x5²

                   =    2⁶ x2² x5⁴ x5²   

                   =    2⁶⁺² x5⁴⁺²

                   =    2⁸ x5⁶       

               

ii)           Solution:

                               6^10-:-6⁵

                         =   6x6x6x6x6x6x6x6x6x6

                              -----------------------------

                               6x6x6x6x6

                            =  6⁵

OR                      6^{10 -5}

                                      = 6⁵. 

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