Worksheet: Exponential Form of a Complex Number

In this worksheet, we will practice converting a complex number from the algebraic to the exponential form (Euler's form) and vice versa.

Q1:

Put 𝑧=53𝑒 in algebraic form.

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

Q2:

Express 𝑍=8 in exponential form.

  • A 𝑒
  • B 8 𝑒
  • C 𝑒
  • D 8 𝑒

Q3:

Express 8𝑖 in exponential form.

  • A 8 𝑒
  • B 𝑒
  • C 8 𝑒
  • D 𝑒

Q4:

Put the number 𝑧=522562𝑖 in exponential form.

  • A 𝑧 = 𝑒
  • B 𝑧 = 5 2 𝑒
  • C 𝑧 = 2 1 0 𝑒
  • D 𝑧 = 5 2 𝑒
  • E 𝑧 = 5 2 𝑒

Q5:

Given that 𝑍=𝑒, find the algebraic form of 𝑍.

  • A 𝑍 = 2 2 𝑒 2 2 𝑒 𝑖
  • B 𝑍 = 𝑒 + 2 2 𝑒 𝑖
  • C 𝑍 = 2 2 𝑒 + 𝑒 𝑖
  • D 𝑍 = 2 2 𝑒 + 2 2 𝑒 𝑖

Q6:

Given that 𝑍=2𝑖1𝑖, write 𝑍 in exponential form.

  • A 𝑒
  • B 𝑒
  • C 2 2 𝑒
  • D 2 2 𝑒

Q7:

Express 11𝑖 in exponential form.

  • A 1 2 𝑒
  • B 1 2 𝑒
  • C 1 2 𝑒
  • D 1 2 𝑒

Q8:

Put 𝑧=7𝑒 in algebraic form.

  • A 𝑧 = 7 𝑖
  • B 𝑧 = 𝑖
  • C 𝑧 = 7 𝑖
  • D 𝑧 = 𝑖

Q9:

Given that 𝑍=𝑒, find the algebraic form of 𝑍.

  • A 𝑍 = 2 2 𝑒 2 2 𝑒 𝑖
  • B 𝑍 = 𝑒 1 2 𝑒 𝑖
  • C 𝑍 = 3 2 𝑒 + 1 2 𝑒 𝑖
  • D 𝑍 = 3 2 𝑒 1 2 𝑒 𝑖

Q10:

Express 𝑍=6 in exponential form.

  • A 𝑒
  • B 6 𝑒
  • C 𝑒
  • D 6 𝑒

Q11:

Express 𝑍=2 in exponential form.

  • A 𝑒
  • B 2 𝑒
  • C 𝑒
  • D 2 𝑒

Q12:

Put the number 𝑧=3232𝑖 in exponential form.

  • A 𝑧 = 𝑒
  • B 𝑧 = 6 𝑒
  • C 𝑧 = 1 6 𝑒
  • D 𝑧 = 6 𝑒
  • E 𝑧 = 6 𝑒

Q13:

Put the number 𝑧=33𝑖 in exponential form.

  • A 𝑧 = 𝑒
  • B 𝑧 = 2 3 𝑒
  • C 𝑧 = 3 6 𝑒
  • D 𝑧 = 2 3 𝑒
  • E 𝑧 = 2 3 𝑒

Q14:

Put the number 𝑧=322𝑖 in exponential form.

  • A 𝑧 = 𝑒
  • B 𝑧 = 2 3 𝑒
  • C 𝑧 = 3 2 2 𝑒
  • D 𝑧 = 3 2 2 𝑒

Q15:

Put the number 𝑧=924+964𝑖 in exponential form.

  • A 𝑧 = 𝑒
  • B 𝑧 = 9 2 2 𝑒
  • C 𝑧 = 2 9 𝑒
  • D 𝑧 = 9 2 2 𝑒
  • E 𝑧 = 9 2 2 𝑒

Q16:

Put the number 𝑧=33𝑖 in exponential form.

  • A 𝑧 = 𝑒
  • B 𝑧 = 2 3 𝑒
  • C 𝑧 = 3 6 𝑒
  • D 𝑧 = 2 3 𝑒
  • E 𝑧 = 2 3 𝑒

Q17:

Express 4𝑖 in exponential form.

  • A 4 𝑒
  • B 𝑒
  • C 4 𝑒
  • D 𝑒

Q18:

Find the numerical value of 𝑒+𝑒.

  • A 3
  • B 3 2
  • C0
  • D 3

Q19:

Given that 𝑎𝑒+𝑏𝑒=(2𝜃)5𝑖(2𝜃)cossin, where 𝑎 and 𝑏, find 𝑎 and 𝑏.

  • A 𝑎 = 2 , 𝑏 = 3
  • B 𝑎 = 2 , 𝑏 = 3
  • C 𝑎 = 2 , 𝑏 = 1
  • D 𝑎 = 2 , 𝑏 = 1

Q20:

Put 𝑧=435𝜋6𝑖5𝜋6cossin in exponential form.

  • A 𝑒
  • B 4 3 𝑒
  • C 𝑒
  • D 3 1 2 𝑒
  • E 4 3 𝑒

Q21:

Put 𝑧=435𝜋6+𝑖5𝜋6sincos in exponential form.

  • A 4 3 𝑒
  • B 1 2 𝑒
  • C 4 3 𝑒
  • D 𝑒
  • E 4 3 𝑒

Q22:

Express the complex number 𝑍=𝑒 in exponential form.

  • A 𝑒 𝑒
  • B 𝑒
  • C 𝑒 𝑒
  • D 𝑒 𝑒

Q23:

Put 𝑧=6𝜋4+𝑖𝜋4cossin in exponential form.

  • A 𝑒
  • B 6 𝑒
  • C 6 𝑒
  • D 2 2 𝑒
  • E 𝑒

Q24:

Given that 𝑧=23+2𝑖 and 𝑧=223𝑖, find 𝑧𝑧, giving your answer in exponential form.

  • A 𝑧 𝑧 = 1 6 𝑒
  • B 𝑧 𝑧 = 1 6 𝑒
  • C 𝑧 𝑧 = 4 𝑒
  • D 𝑧 𝑧 = 1 6 𝑒

Q25:

Given that 𝑧=2(90𝑖90)cossin and 𝑧=4(30+𝑖30)sincos, find 𝑧𝑧, giving your answer in exponential form.

  • A 𝑧 𝑧 = 8 𝑒
  • B 𝑧 𝑧 = 8 𝑒
  • C 𝑧 𝑧 = 8 𝑒
  • D 𝑧 𝑧 = 8 𝑒
  • E 𝑧 𝑧 = 6 𝑒

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