express the following complex numbers in the standard form of a complex number

Question

u=2-3i and 

v=-3+i

Write the following complex numbers in the standard form of a complex number.

u+v=

u-v=

uv=

u/v=

v/u=

v^2=

v^3=

v^4=

 

Answer 1

U+ v = 2-3i -3 + i = -1 -2i

 

U- V = ( 2 - 3 i) - ( -3 + i) = 5 - 4i

 

U V = ( 2- 3i) ( -3 +i) =- 6 + 9i + 2i +3 = -3 + 11i

 

U/ V = ( 2 - 3i)/( -3 + i ) *  ( -3  - i) / ( -3 - i) =

     

            ( - 6 + 9i -2i -3i ) / 10=

 

             - 6/10 + 6/10i

V/u  / Do it yourself 

 

V^2 = ( - 3 + i) = 9 - 1 -6i = 8 - 6i

 

 V ^3 = ( - 3 + i ) ^3 = ( -3) ^3 +3(-3)^2 i + 3 ( -3) i^2 +i^3

 

                                = -27 -27i +9 - i = - 18 - 28 i

V^4 = (v^2 ) ^2 = ( 8-6i) ^2 = 64 -96i +36 = 100 -96i

 

 

 

 

 

 

Answer 2

Hi Dalia,

 

u=2-3i and v=-3+i

 

u+v=(2-3i)+(-3+i)

      =2-3-3i+i

      =-1-2i

 

u-v=(2-3i)-(-3+i)

     =2-3i+3-i

     =5-4i

 

uv=(2-3i)(-3+i)

   =2(-3+i)-3i(-3+i)

   =-6+2i+9i-3i²

   =-6+11i+3       (i²=-1)

   =-3+11i

 

u/v=(2-3i)/(-3+i)

 

multiply the numerator and denominator by (-3-i)

 

u/v=(2-3i)(-3-i)/(-3+i)(-3-i)

     =(-6-2i+9i+3i²)/ (-3)²-(i)²

 

    =(-6+7i-3)/9+1

 

    =-9+7i/10

 

   =-9/10+7i/10

 

v/u=(-3+i)/(2-3i)

 

Multiply the numerator and denominator by 2+3i

 

v/u=(-3+i)(2+3i)/(2-3i)(2+3i)

 

    =(-6-9i+2i+3i²)/(2)²-(3i)²

    =(-6-7i-3)/4+9

 

   =-9-7i/13

 

   =-9/13-7i/13

 

v²=(-3+i)²

   =(-3)²+2(-3)(i)+(i)²

 

   =9-6i-1

   =8-6i         equation 1

 

v³=(-3+i)³

   =(-3)³+3(-3)²(i)+3(-3)(i)²+(i)³

   =-27+27i+9-i

   =-18+26i

 

v⁴=(-3+i)⁴    

   =((-3+i)²)²

  

 =(8-6i)²   from equation 1

 

 =(8)²-2(8)(6i)+(6i)²

=64-96i+36i²

 =64-96i-36

=28-96i

=4(7-24i)