IBPS - QUANTITATIVE APTITUDE - SURDS AND INDICES

SURDS AND INDICES

Meaning of Power and Indices :

       We know very well that -----

        = 2 * 2 = 4 = 2 is multiplied two times by 2.

        = 2 * 2 * 2 = 8 = 2 is multiplied three times by 2.

        = 4 * 4 * 4 = 64 = 4 is multiplied three times by 4.

        Thus , we can say, that

         = a * a* a*................ till 'm' factors. It means 'a' is multiplied 'm' times by 'a', where 'a' is real number and 'm' is a positive integer.

         Now, the above mathematical expression is defined below on the basis of power and indices----

         = a * a* a*................ till 'm' factors. Here, 'a' is called base, 'm' is called index an '' is called m th power of base 'a'. Thus. it becomes clear that is product which is obtained from multiplying 'a' m times by 'a'.

          Now , we can take an example for clarifying above mentioned definition.

          = 2 * 2 * 2 = 8 

          here,

          2 = base

          3 = index

          = 3rd power of base 2.

Fundamental Rules of Indices :

Rule I  * =

Example (I)  * = ?

(1)    (2)    (3)    (4)    ( 5) None

Explanation:

  * =  

   * =   =    // because here a=2,m=3 & n=5

Example (2) * = ?

(1)    (2)    (3)    (4)    ( 5) None

Explanation

* =

*= =     //here a=9, m=4 and n=-2  

Example (3)  * = ?

  

(1) 9   (2)    (3)    (4)    ( 5) None

 Explanation

* =

  * =   =  = = 9

Remarkable point

In the formula * = , Base 'a' =a real number. 'm' and 'n' may be positive, or negative integers or any fraction.

Rule II  

  

Example (1)  = ?

(1)   (2)    (3)   ( 4) None

Explanation

 

 = =    //here m=2 and n=3

Example (2)  = ?

(1)   (2)    (3)    (4)    ( 5) None  

Explanation

 

  = =   // here a=5, m=4 and n=3

Example (3)   = ?

(1)   (2)    (3)    (4)    ( 5) None  

Explanation

 

  = = =   // here a=2, m=3 and n=

Remarkable Point

Example 

 = =

but,   = =

since, 

therefore, 

Rule III    or  

Example (1)  = ?

(1)   (2)    (3)    (4)    ( 5) None  

Explanation

=                       // here a=8 and m=

   = 

      = 

      = 

Example (2) = ?

(1)   (2)    (3)    (4)    ( 5) None  

Explanation :

  =                // Applying rule No. III

         =              // 27=3 * 3 *3 =

           =                // Applying rule No. II

        =  = 

Rule IV  

Example (1)   = ?
 

 (1)   (2)    (3)    (4)    ( 5) None 

Explanation 

  = =           // Applying Rule No. II

 =                         // Applying Rule No. IV

 =

Example (2)   = ?

 (1)   (2)     (3)    (4)     ( 5) None  

Explanation :

   =   

                  =              // Applying Rule No. IV

                  =  

                  =               // Applying Rule No. III

Rule V    

Example (1)  = ?

 (1) 4   (2)  8    (3) 16    (4)  2    ( 5) None 

Explanation : 

First Method :  
                          =                 // Applying Rule No. V

                                    = = 2

Second Method :

                          =

                                    =                    // Applying Rule No. II

                                    = 2

Example (2)   = ?

 (1) 16   (2)  8    (3)  4    (4)  2   ( 5) None

Explanation : 

       =                           //Applying Rule No. V
            =  =
            = = 4

Example (3)   = ?

 (1) 5   (2)  25    (3)     (4)  -125   ( 5) None

Explanation : 

=                   // Applying Rule No. III
               =
               =                        // Applying Rule No. V
               =  

Rule VI 
Where 'a' and 'b' both are real numbers.

Example (1)  = ?
 (1)   (2)     (3)     (4)    ( 5) None

Explanation : 

 

  =              // here a=3, b=4 and m=2 

          =  

Example (2)  = ? * 

(1)   (2)    (3)     (4)    ( 5) None

Explanation :

  =  

            =                      //Applying Rule No. VI

            = 

Rule VII    

Example (1)    

(1) 9   (2) 6    (3)  2    (4)  3   ( 5) None

Explanation :

        =             // here a=9, b=6 and m=4 

             =  

So, correct answer will be (4) 

Example (2)  

 (1) 625   (2) 64    (3)  100    (4)  225   ( 5) None

Explanation :  

        =             // Applying Rule No. VII

                     =     

                     =    = 

So, correct answer will be (2) 

Rule VIII   , where  

Example (1)    = ?




(1)   (2)     (3)     (4)  1   ( 5) None

Explanation :  

  =                 // Applying Rule No. I

              =  

              =  

              = 1

Example (2)    = ?
(1) 0   (2) 1    (3) 26     (4)  8   ( 5) None

Explanation :   

Since  

Thus ,  = 1