Page 43 - WCS - Electrical
P. 43

28+
                       7
                      +
                    =
                 28+
                       7
                      +
                  4 =
                 56 4
                + +
                       14
                    =
                 56
                +
                       14
                  4 =
                 -
                      −
                 72 4
                --
                    =
                     −8
                 72
                -
                      −8
                  9 =
                +
                − 96 9
                +
                    =
                      +16
                − 96
                      +16
                5y − 6
                     20
                   = 20
                5y
                     5
                5 =
                     5
                x 5
                x
                4
                5x 4
                5x
                   = 25
                     5
                5 =
                5
                     5
                10a
                10a
                2a
                2a
                 8ac
                − 8ac
                 2bc
                −
                − 2bc
                 5m −×
                       6n −
                           7p
                       6n −
                −
                 5m −×
                     28mn
                   −
                5a +−
                    20 28mn
                5a +
                    20
                    28
                7a +
                7
                 m 28a +
                a
                 m
                     a
                a
                   =
                      m−
                     a
                 n =
                a
                 n
                 3 a

                2
                      3
                 3
                     2 −
                            2 =
                   =
                         2 =
                2
                             1
                      3
                                2
                         2 =
                 2 =
                            2 =
                     2 −
                2 + − 6 = 25 − m− n n  7p 1  2
                 2
                2
                2× 2× 2  8
                2× 2× 2  = 8  = 2
                             2
                    2
                         4 =
                       =
                 2×
                    2
                        n 4
                 n 2×
          WORKSHOP CALCULATION & SCIENCE  - CITS
                a
                    
                     a  n
                     a
                a n  =    
                b n =    b  
                 n
                      b 
                        2
                2 b 3    2  2  2× 2  4
                 3
               (2 x 2) x (2 x 2) x (2 x 2) = (2 x 2 x 2) (2 x 2 x 2)
                          = 2×
                      2
                2
                                = 4
                               2
                   = 
                        
                3 2 =   3    =  3× 3 =  9
                 2
                      3 
                     
               4 x 4 x 4 = 64 3× 3  9
                      1
                3
                 −
                  n
                a  − n  = 1
                    =
                       n
                a
               8 x 8 = 64
                     a
                       n
                      1 a
                 −
                  2
                2
                    = 1
           A mixed number raised to a power is first converted into an improper fraction and then the result is evaluated.
                  2
                 −
                       2
                2
                    =
                     2
                    2 2
                           2
                  3  2  2   7  2
                        7
                 3   =    
                1  
                
                 1   4    =   4  
                  4
                7   × 7 = 49 4
                       49 
                   7 
                        
                7
                4 × 4 = 16
                4  4   16
           Indices
           •  The indices are added in multiplication
                     m  x  a = a m+ n                                 2  3    2
                         n
                   a
                         n
                     m  x  a = a m+ n                             =    3x 2 y 3  =  y 2
                   a
                              m+
                   a
                     m m
                                 n n
                              m+
                                                                             y
                         n n
                    a
                        a = =
                                                                    3x 2 2
                                                                        53 3
           •  The indices are subtracted in division                =    − 3x 6xy yx   y y = 2 2x y y 3 3 2 2  2

                       x  a x  m
                                   m+
                             a a n
                                     n
                                                                    3
                             a =
                                 a
                        a
                            x
                                                                               3
                     m    m−                                      = =  −  6x 5 5 = 5 = = y3x  2x 3 3 =  y
                                                                         y
                    a
                         a   m
                             n n
                      m  = a   x  a = a m+ n                        − 6x− 6x    y y  2x  32
                                                                             2x 32
                                                                             5
                    a
                      a a   a n m m  =   m  n                            = − 6x 3x  y y  = 2xy
                           m−
                             n
                         a m− nm−
                      n = a=a
                    a n n  a  = a m− n                                     − 6x 5 y  2x 3
                    a a  aa                                         3x 3 y 2  2
                          n   m
                                                                 =
                                    n
                                 m−
                                                                        2
                                                                     3
                            m.
                    m
                             n a=
                       n
           •  In case of index of an index, both the indices are multiplied mutually =  3x  2 y
                          a
                   [
                   a ] =
                                                                   3x 3
                                                                       y 2

                                                                     xy 23
                            n
                     m  n  a m. n                                = 3x y3x   y  = 3 3x 2 2 y
                                                                            2
                   a ] =
                                                                           3x
                                                                               y y
                       [ m nm  n    a m. nm. n  m. n             = =  xy 3x = 3x= y  = 3x 2 y
                   [ a ][
                                                                     xy xy =
                   a ] = = m
                           a a n
                        a ] =
                               a
                        [

                                                                          xy 3
                         m
                    1/m
                                                                         2
           •  A fractional index shows root of a number                45a =  3x 2 c y 2  = 3x 2 y
                  a
                       =
                                  m.
                                                                          b
                              n
                           a m
                                    n
                    1/m  m a ][  =  a                               =     2  2 xy
                                                                           b
                  a 1/m
                      1/m   =  m m a                                    = 45 9a ba ba    2 2 2 c  2
                                                                      45a 2 2
                  a a
                                                                      45
                       = = 1/m
                           a a m
                                                                           c c c 2
                       a
                                a
                                                                                  c
                                                                               b
                                                                          2 45a
                            =
                                                                    = =
                                                                           c
                                                                         =
                                                                        9a 2 2
           •  In case of an index having minus sign, the sign can be changed by taking the number from numerator to

                                                                        9a
                                                                           c c
                                                                        9a

                         1   1/m
                               m
                                                                               22
                                                                                  2
                    m
                   −
                      = a
                                 a
                              =
                                                                             45
                  a

              denominator or vice versa                                    =  9aa cb  c
                         1
                    m
                   −
                         m
                  a      − − m m  =  a1 1                                        2
                          m
                   a a  = =   a − a m m=  1 m                                 9a  c
                         m
                        a a
                             a 1
                   1
                          m m
                and      =   a a −  =
                    1 m   m     m
                   −
                    a   1 1 m = a=   1  a
                       =
                         a m m
                         a
                   −
                   − −
                  a m m     = a m
                  a a
                        −
                         m
                  a m /  n  = a  1  = a m
                        n m
                         a

                  m /
                     n
                        n m m
                          −
           •  If an index contains both the numerator and denominator then it means that the number has ‘index’ as well as
                         a
                 a
                      = n n
                  m / /
                     n n
                  m
                         a m m

                 a a
              ‘root’. 2 a = = m / 2 =  n m
                         a a n
                              a


                   3x +  2y m /  n m
                   2
                         2 n
                        2
                      2y
                   2 + a
                     2x +  3y 2 2  =  a
                 3x 2 2
                 3x +
                 3x
                      2y 2y
                   2 +
                 2x +3x
                 _______ +
           Basic problem22 2  3y 2 2 2  2y 2
                      3y
                 2x + +
                 2x
                  2 _______
                       2 3y 2

                 x + 5y 2x + 2  3y  2  2
                 _______
                 _______
           Addition 2   2 3x + 2y
                 x +
                        2 2
                 _______ 22 2  5y _______  2
                             3y
                     5y 5y2x +
                 x + +
                 x
                  2 _______ 2
                            2
           1   5x y + 3xy  + 8x y + 7xy 2
                      x +
                         2
                          5y
                              2
                 _______ _______
                 _______
                      _______
                         2 2
                              2
                       x +
               = 5x y + 8x y + 3xy   + 7xy 2
                            5y
                                2
                   2
                  12x 3 y _______
                       2
                         =

                    3
                     2 2
                          3xy
               =  13x y + 10xy 2
                 12x
                      y

                    2 23 23
                          3xy
                 12x
                         =
                  4x
                      y y y
                 12x
                         3
                            2
                      3 2
                          3xy
                      12x
                         = 3xy= y
           2   Add 5a , + 12b , - c , + a , - 4b , + 3
                                          3
                                 3
                             3
                                     3
                      y
                              =
                               3xy
                  4x 2 2
                  4x  y y    2 3  2
                   4x
               5a  + 12b  + (- c ) + a  + (- 4b ) + 3
                                  3
                 15y 3 15  3 4x 12x y y  3  =  3xy  3
                        10 2
                   3 =
                         3 4x
                     15y 15    = y y 10  3 y
               = 6a  + 8b  - c  + 3
                        10
                   5 15
                   15
               15y
               15y
               15y
                       15
                        10
                   5 = y=
                15y
           Subtract5 5 15y y 515 =  y 10
                15y
                15y

                    15yy
           1    3a 2  x  4a 15 5a 2 3 y=  2 10  2  2
               Subtract 2x  - 3y  from 3x  + 2y
                       x
                 2  x  4a 15y 5 3
                       x
                  3a 2 2  4 4     x  4a 2 10a 5a x  5a x  5a 3 3  3
               3a
                 6a 4a x   x
               3a
                        x
                   3a
                 6a 4 4  x  10a 4a  x  5a
                 6a
                      10a
                 6a
                      x  10a x  4 2
                   60a   3 6 6a  x a   x  4 10a5a x a  3
                      6
                  60a 6a
                    =   = 60a 5 6 6  = = a 4  x  10a
                         a
                  60a
                  60a
                         a
                = =    5 = a= 60a 6                        30
                  60a 5 5 =  = a
                  60a    60a 60a
                  60a
                          5 6
                  − 25a 15 =  5  = a        CITS : WCS - Electrical - Exercise 4
                =      1560a
                  −
                       8 15
                    = − 25a− 25a −15
                  −25a
                   5a

                      − −
                = =  −  5a 8− 8− 8 25a 15
                     =
                            815
                           −
               4x 2 y − 5a− 5a   −− 5a 25a
             =    2  =  2x = 2
                        2
               4x 2
                 = 4x y4x  y  =  2x 2 2 − 5a − 8
                2y 2
                  y

                      2
             = =  2y = 2x= y4x 2x  2
                2y=    2y 2 =  2x
                2y
                   =  4x  y  =  2x 2
                      2y
   38   39   40   41   42   43   44   45   46   47   48