Worksheet: Dividing Complex Numbers

In this worksheet, we will practice performing division on complex numbers.

Q1:

Simplify 2 3 + 𝑖 .

  • A 2 3 2 𝑖
  • B ( 3 𝑖 ) 1 0
  • C ( 3 𝑖 ) 5
  • D 3 𝑖
  • E 2 3 + 2 𝑖

Q2:

Put 1 8 9 𝑖 3 𝑖 in the form 𝑎 + 𝑏 𝑖 .

  • A 2 7 5 4 𝑖
  • B 9 + 1 8 𝑖
  • C 3 + 5 4 𝑖
  • D 3 + 6 𝑖

Q3:

Simplify 3 6 𝑖 1 5 𝑖 .

  • A 1 1 8 3 8 𝑖
  • B 3 + 6 5 𝑖
  • C 3 3 2 6 + 9 2 6 𝑖
  • D 9 1 3 5 2 6 𝑖
  • E 3 4 + 5 2 4 𝑖

Q4:

Simplify 4 + 𝑖 4 𝑖 .

  • A 1 𝑖
  • B 1 5 + 8 𝑖 1 5
  • C 1 7 1 5 + 8 𝑖
  • D 1 5 + 8 𝑖 1 7
  • E 1 7 + 8 𝑖 1 5

Q5:

Express ( 5 + 5 𝑖 ) ( 8 4 𝑖 ) 1 𝑖 in the form 𝑎 + 𝑏 𝑖 .

  • A 2 0 + 4 0 𝑖
  • B 2 8 + 1 6 𝑖
  • C 6 0 + 2 0 𝑖
  • D 4 0 2 0 𝑖
  • E 5 + 4 0 𝑖

Q6:

Simplify 1 2 4 𝑖 2 𝑖 .

  • A 8 2 4 𝑖
  • B 4 + 1 2 𝑖
  • C 2 + 2 4 𝑖
  • D 2 + 6 𝑖

Q7:

Put 2 5 + 5 𝑖 in the form 𝑎 + 𝑏 𝑖 .

  • A 0 . 2 0 . 2 𝑖
  • B 0 . 2 + 0 . 2 𝑖
  • C 0 . 2 + 0 . 2 𝑖
  • D 0 . 2 0 . 2 𝑖

Q8:

Simplify 1 3 1 3 + 2 𝑖 .

  • A 1 6 9 1 7 3 2 6 1 7 3 𝑖
  • B 1 6 9 1 7 3 + 2 6 1 7 3 𝑖
  • C 1 6 9 1 6 5 2 6 1 6 5 𝑖
  • D 1 6 9 1 6 5 + 2 6 1 6 5 𝑖

Q9:

Simplify ( 8 + 7 𝑖 ) ( 1 + 8 𝑖 ) ( 6 + 4 𝑖 ) .

  • A 1 5 + 3 0 9 1 0 𝑖
  • B 1 1 3 3 0 9 2 6 𝑖
  • C 1 5 3 0 9 1 0 𝑖
  • D 1 1 3 + 3 0 9 2 6 𝑖

Q10:

Simplify ( 3 + 2 𝑖 ) ( 3 + 3 𝑖 ) ( 4 + 𝑖 ) ( 4 + 4 𝑖 ) .

  • A 1 5 3 4 3 3 6 8 𝑖
  • B 1 5 3 4 + 3 3 6 8 𝑖
  • C 1 5 3 4 + 3 3 6 8 𝑖
  • D 1 5 3 4 3 3 6 8 𝑖

Q11:

Simplify 1 9 𝑖 + 5 𝑖 7 𝑖 6 + 4 𝑖 4 𝑖 + 2 𝑖 .

  • A 1 𝑖
  • B 1 2 𝑖
  • C 1 + 𝑖
  • D 1 + 2 𝑖

Q12:

Simplify 7 + 𝑖 + 3 𝑖 6 𝑖 2 𝑖 𝑖 𝑖 .

  • A 1 0 + 7 𝑖
  • B 4 3 𝑖
  • C 1 0 7 𝑖
  • D 4 + 3 𝑖

Q13:

Solve the equation 𝑖 𝑧 = 4 + 3 𝑖 .

  • A 𝑧 = 4 + 3 𝑖
  • B 𝑧 = 3 4 𝑖
  • C 𝑧 = 3 + 4 𝑖
  • D 𝑧 = 3 + 4 𝑖
  • E 𝑧 = 3 4 𝑖

Q14:

If 𝑎 + 𝑏 𝑖 = 3 5 𝑖 3 + 5 𝑖 , is it true that 𝑎 + 𝑏 = 1 ?

  • Ayes
  • Bno

Q15:

Find the real values of 𝑥 and 𝑦 that satisfy the equation 7 9 𝑖 1 2 𝑖 = 𝑥 + 𝑦 𝑖 .

  • A 𝑥 = 5 , 𝑦 = 1
  • B 𝑥 = 6 3 5 , 𝑦 = 2 5
  • C 𝑥 = 7 , 𝑦 = 9 2
  • D 𝑥 = 2 1 , 𝑦 = 2 3
  • E 𝑥 = 2 5 3 , 𝑦 = 5 3

Q16:

Put 4 3 + 3 2 3 8 in the form 𝑎 + 𝑏 𝑖 .

  • A 4 1 1 1 2 1 1 𝑖
  • B 4 1 1 + 1 2 6 1 1 𝑖
  • C 4 1 1 + 1 2 1 1 𝑖
  • D 4 1 1 1 2 6 1 1 𝑖
  • E 4 1 1 + 1 2 6 𝑖

Q17:

By what complex number would you multiply the numerator and denominator of the quotient 5 + 6 𝑖 3 2 𝑖 in order to simplify it?

  • A 3 2 𝑖
  • B 5 + 6 𝑖
  • C 3 2 𝑖
  • D 5 6 𝑖
  • E 3 + 2 𝑖

Q18:

Simplify 7 ( 4 + 𝑖 ) .

  • A 1 0 5 2 8 9 5 6 2 8 9 𝑖
  • B 1 0 5 2 8 9 5 6 2 8 9 𝑖
  • C 1 0 5 2 4 1 + 2 8 2 4 1 𝑖
  • D 1 0 5 2 8 9 + 5 6 2 8 9 𝑖

Q19:

Expand and simplify ( 𝑝 + 𝑞 𝑖 ) ( 𝑝 𝑞 𝑖 ) .

  • A ( 𝑝 + 𝑞 )
  • B 𝑝 + 𝑞
  • C ( 𝑝 𝑞 )
  • D 𝑝 𝑞
  • E 𝑞 𝑝

Expand ( 𝑎 + 𝑏 𝑖 ) ( 𝑝 𝑞 𝑖 ) .

  • A ( 𝑎 𝑝 𝑏 𝑞 ) + ( 𝑏 𝑝 𝑎 𝑞 ) 𝑖
  • B ( 𝑎 𝑝 𝑏 𝑞 ) ( 𝑏 𝑝 𝑎 𝑞 ) 𝑖
  • C ( 𝑎 𝑝 + 𝑏 𝑞 ) + ( 𝑏 𝑝 𝑎 𝑞 ) 𝑖
  • D ( 𝑎 𝑝 + 𝑏 𝑞 ) ( 𝑏 𝑝 𝑎 𝑞 ) 𝑖
  • E ( 𝑎 𝑞 + 𝑏 𝑝 ) + ( 𝑏 𝑞 𝑎 𝑝 ) 𝑖

Hence, find a fraction which is equivalent to 𝑎 + 𝑏 𝑖 𝑝 + 𝑞 𝑖 and whose denominator is real.

  • A ( 𝑎 𝑝 + 𝑏 𝑞 ) ( 𝑏 𝑝 𝑎 𝑞 ) 𝑖 𝑝 𝑞
  • B ( 𝑎 𝑞 + 𝑏 𝑝 ) + ( 𝑏 𝑞 𝑎 𝑝 ) 𝑖 𝑝 + 𝑞
  • C ( 𝑎 𝑝 + 𝑏 𝑞 ) ( 𝑏 𝑝 𝑎 𝑞 ) 𝑖 𝑝 + 𝑞
  • D ( 𝑎 𝑝 + 𝑏 𝑞 ) + ( 𝑏 𝑝 𝑎 𝑞 ) 𝑖 𝑝 + 𝑞
  • E ( 𝑎 𝑝 + 𝑏 𝑞 ) + ( 𝑏 𝑝 𝑎 𝑞 ) 𝑖 𝑝 𝑞

Q20:

Simplify 2 + 4 𝑖 𝑖 .

  • A 2 + 4 𝑖
  • B 2 4 𝑖
  • C 4 + 2 𝑖
  • D 4 2 𝑖
  • E 4 2 𝑖

Q21:

Solve the equation 𝑧 ( 2 + 𝑖 ) = 3 𝑖 for 𝑧 .

  • A 𝑧 = 3 2 𝑖
  • B 𝑧 = 1 + 𝑖
  • C 𝑧 = 1 + 𝑖 2
  • D 𝑧 = 1 𝑖 2
  • E 𝑧 = 1 𝑖

Q22:

Put 1 + 2 𝑖 9 + 7 𝑖 in the form 𝑥 + 𝑦 𝑖 .

  • A 2 3 1 3 0 1 1 1 3 0 𝑖
  • B 2 3 1 3 0 + 1 1 1 3 0 𝑖
  • C 2 3 3 2 1 1 3 2 𝑖
  • D 2 3 3 2 + 1 1 3 2 𝑖

Q23:

Put 1 2 𝑖 ( 1 + 𝑖 ) in the form 𝑥 + 𝑦 𝑖 .

Q24:

Simplify 2 2 𝑖 3 𝑖 .

  • A 4 2 𝑖 5
  • B 2 4 𝑖 5
  • C 1 2 + 4 𝑖 5
  • D 2 3 2 𝑖
  • E 2 2 𝑖 5

Q25:

Express 1 ( 3 + 𝑖 ) in the form 𝑎 + 𝑏 𝑖 .

  • A 4 5 + 3 5 𝑖
  • B 2 2 5 + 3 5 0 𝑖
  • C 2 2 5 + 3 5 0 𝑖
  • D 2 2 5 3 5 0 𝑖
  • E 4 5 3 5 𝑖

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.