Simplify each of the following:

Question

7)  Simplify each of the following:
 
a)√-90

b)(4-3i)(5+2i)

c)i^59

d)(5-3i) with a square outside the bracket.

e)2+5i/3-2i

 

Answer 1

a)

√(90)(-1)

=√(90)(i^2)                          (Since  -1 = i^2)

=√(90)√(i^2)

=√((9)(10))i

=3√10 i

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

b)

(4-3i)(5+2i)

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

= 20+8i-15i-(3i)(2i)

=20 - 7i - 6(i)^2

=20 - 7i - 6(-1)                           (Since  i^2 = -1)

=20 - 7i +6

=26-7i    

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

c)

i^59 = i^(58+1)

      =i^(58)*i

    =((i^2)^29)*i

   =(-1)^29*i                            (Since  -1 = i^2)

  =-1*i

 =-i

Answer 2

d)

(5 - 3i)^2

Use formula: (a - b)^2 = a^2 -2ab + b^2

(5 - 3i)^2 = (5)^2- 2(5)(3i) + (3i)^2

             = 25 -30i + 9i^2

             = 25 -30i - 9

             =16 - 30i

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

e)

2+5i/3-2i

Multiply and divide with 3+2i

= (2+5i)(3 +2i )/( 3-2i )(3 + 2i)

= (6 + 4i +15i +10i^2)/(3^2-(2i)^2)

= (6 + 19i -10)/(9-4i^2)

=(-4 + 19i )/(9+4)

=(-4 + 19i )/13