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=4√3ο€Ό5πœ‹6βˆ’π‘–5πœ‹6cossin in exponential form.

  • Aπ‘’οŽ¦ο‘½οŽ₯
  • B4√3π‘’οŽ€ο‘½οŽ₯
  • Cπ‘’οŽ€ο‘½οŽ₯
  • D√312π‘’οŽ¦ο‘½οŽ₯
  • E4√3π‘’οŽ¦ο‘½οŽ₯

Q2:

Express the complex number 𝑍=π‘’οŠ±οŠͺοŠ±οƒοŽ‘οŽ’ο‘½οŽ οŽ‘ in the form of π‘Ÿβ‹…π‘’οΌοƒ.

  • A𝑒⋅𝑒οŠͺοƒο‘½οŽ οŽ‘
  • Bπ‘’οŠ±οƒοŽ¦οŽ ο‘½οŽ οŽ‘
  • Cπ‘’β‹…π‘’οŠ±οŠͺοƒο‘½οŽ οŽ‘
  • Dπ‘’βˆ’π‘’οŠ±οŠͺοƒο‘½οŽ οŽ‘

Q3:

Given that 𝑧=2(90βˆ’π‘–90)∘∘cossin and 𝑧=4(30+𝑖30)∘∘sincos, find π‘§π‘§οŠ§οŠ¨, giving your answer in exponential form.

  • A𝑧𝑧=8π‘’οŠ§οŠ¨οƒοŽ οŽ ο‘½οŽ₯
  • B𝑧𝑧=8π‘’οŠ§οŠ¨οƒοŽ€ο‘½οŽ₯
  • C𝑧𝑧=8π‘’οŠ§οŠ¨οƒοŽ€ο‘½οŽ’
  • D𝑧𝑧=8π‘’οŠ§οŠ¨οƒοŽ‘ο‘½οŽ’
  • E𝑧𝑧=6π‘’οŠ§οŠ¨οƒοŽ οŽ ο‘½οŽ₯

Q4:

Put the number 𝑧=5√22βˆ’5√62𝑖 in exponential form.

  • A𝑧=π‘’οŽ€ο‘½οŽ’οƒ
  • B𝑧=5√2π‘’οŽ οŽ ο‘½οŽ₯
  • C𝑧=√210π‘’οŽ€ο‘½οŽ’οƒ
  • D𝑧=5√2π‘’οŽ€ο‘½οŽ’οƒ
  • E𝑧=5√2π‘’οŽ‘ο‘½οŽ’οƒ

Q5:

Given that 𝑍=√2𝑖1βˆ’π‘–, write 𝑍 in exponential form.

  • Aπ‘’οŽ’ο‘½οŽ£οƒ
  • Bπ‘’οŠ±οƒοŽ’ο‘½οŽ£
  • C√22π‘’οŠ±οƒοŽ’ο‘½οŽ£
  • D√22π‘’οŽ’ο‘½οŽ£οƒ

Q6:

Given that 𝑧=3ο€Ό11πœ‹6+𝑖11πœ‹6cossin, find 1𝑧 in exponential form.

  • A1𝑧=13𝑒οŽ₯
  • B1𝑧=3π‘’οŽ οŽ ο‘½οŽ₯
  • C1𝑧=13π‘’οŽ οŽ ο‘½οŽ₯
  • D1𝑧=𝑒οŽ₯

Q7:

Given that 𝑧=12βˆ’βˆš32π‘–οŠ§ and 𝑧=2√3+2π‘–οŠ¨, find π‘§π‘§οŠ§οŠ¨, giving your answer in exponential form.

  • A𝑧𝑧=4π‘’οŠ§οŠ¨οƒοŽ’ο‘½οŽ‘
  • B𝑧𝑧=π‘’οŠ§οŠ¨οƒοŽ’ο‘½οŽ‘
  • C𝑧𝑧=14π‘’οŠ§οŠ¨οƒο‘½οŽ₯
  • D𝑧𝑧=14π‘’οŠ§οŠ¨οƒοŽ’ο‘½οŽ‘
  • E𝑧𝑧=14π‘’οŠ§οŠ¨οƒοŽ€ο‘½οŽ’

Q8:

Given that 𝑧=5π‘’οŠ§οŠ±ο‘½οΌοŽ‘ and 𝑧=6π‘’οŠ¨ο‘½οΌοŽ’, express π‘§π‘§οŠ§οŠ¨ in the form π‘Ž+𝑏𝑖.

  • A𝑧𝑧=15√3βˆ’15π‘–οŠ§οŠ¨
  • B𝑧𝑧=11√32+112π‘–οŠ§οŠ¨
  • C𝑧𝑧=15βˆ’15√3π‘–οŠ§οŠ¨
  • D𝑧𝑧=15√3+15π‘–οŠ§οŠ¨
  • E𝑧𝑧=11√32βˆ’112π‘–οŠ§οŠ¨

Q9:

Put 𝑧=6ο€»βˆ’πœ‹4+π‘–πœ‹4cossin in exponential form.

  • Aπ‘’ο‘½οŽ£οƒ
  • B6π‘’ο‘½οŽ£οƒ
  • C6π‘’οŽ’ο‘½οŽ£οƒ
  • D√22π‘’οŽ’ο‘½οŽ£οƒ
  • Eπ‘’οŽ’ο‘½οŽ£οƒ

Q10:

Given that π‘Žπ‘’+𝑏𝑒=(2πœƒ)βˆ’5𝑖(2πœƒ)οŠ¨οƒοΌοŠ±οŠ¨οƒοΌcossin, where π‘Žβˆˆβ„ and π‘βˆˆβ„, find π‘Ž and 𝑏.

  • Aπ‘Ž=βˆ’2, 𝑏=3
  • Bπ‘Ž=2, 𝑏=3
  • Cπ‘Ž=βˆ’2, 𝑏=βˆ’1
  • Dπ‘Ž=2, 𝑏=βˆ’1

Q11:

Put 𝑧=5√3π‘’ο‘½οŽ’οƒ in algebraic form.

  • A𝑧=5√32βˆ’152𝑖
  • B𝑧=12+√32𝑖
  • C𝑧=5√32+152𝑖
  • D𝑧=βˆ’12βˆ’βˆš32𝑖
  • E𝑧=βˆ’5√32+152𝑖

Q12:

Given that 𝑧=2√3+2π‘–οŠ§ and 𝑧=βˆ’2βˆ’2√3π‘–οŠ¨, find π‘§π‘§οŠ§οŠ¨, giving your answer in exponential form.

  • A𝑧𝑧=16π‘’οŠ§οŠ¨οƒοŽ’ο‘½οŽ‘
  • B𝑧𝑧=16π‘’οŠ§οŠ¨οƒοŽ£ο‘½οŽ’
  • C𝑧𝑧=4π‘’οŠ§οŠ¨οƒοŽ’ο‘½οŽ‘
  • D𝑧𝑧=16π‘’οŠ§οŠ¨οƒο‘½οŽ₯

Q13:

Put 𝑧=βˆ’4√3ο€Ό5πœ‹6+𝑖5πœ‹6sincos in exponential form.

  • Aβˆ’4√3π‘’οŽ‘ο‘½οŽ’οƒ
  • B12π‘’οŽ‘ο‘½οŽ’οƒ
  • C4√3π‘’οŽ‘ο‘½οŽ’οƒ
  • Dπ‘’οŽ‘ο‘½οŽ’οƒ
  • E4√3π‘’οŽ€ο‘½οŽ₯

Q14:

Given that 𝑍=π‘’οŠ¨οŠ±οƒοŽ€ο‘½οŽ£, find the algebraic form of 𝑍.

  • A𝑍=√22π‘’βˆ’βˆš22π‘’π‘–οŠ¨οŠ¨
  • B𝑍=𝑒+√22π‘’π‘–οŠ¨οŠ¨
  • C𝑍=βˆ’βˆš22𝑒+π‘’π‘–οŠ¨οŠ¨
  • D𝑍=βˆ’βˆš22𝑒+√22π‘’π‘–οŠ¨οŠ¨

Q15:

Express 11βˆ’π‘– in exponential form.

  • A1√2π‘’οŠ±οƒο‘½οŽ£
  • B12π‘’ο‘½οŽ£οƒ
  • C1√2π‘’ο‘½οŽ£οƒ
  • D12π‘’οŠ±οƒο‘½οŽ£

Q16:

Find the numerical value of 𝑒+π‘’οŽ οŽ ο‘½οŽ₯οŽ οŽ ο‘½οŽ₯οƒοŠ±οƒ.

  • A√3
  • B√32
  • C0
  • Dβˆ’βˆš3

Q17:

Express 𝑍=βˆ’8 in exponential form.

  • Aπ‘’οŽ„οƒ
  • B8π‘’οŠ¦οƒ
  • Cπ‘’οŠ¦οƒ
  • D8π‘’οŽ„οƒ

Q18:

Express βˆ’8𝑖 in exponential form.

  • A8π‘’οŠ±οƒο‘½οŽ‘
  • Bπ‘’οŠ±οƒο‘½οŽ‘
  • C8π‘’ο‘½οŽ‘οƒ
  • Dπ‘’ο‘½οŽ‘οƒ

Q19:

Given that 𝑍=ο€Ό2πœ‹3+𝑖2πœ‹3cossin, express π‘βˆ’1 in exponential form.

  • Aπ‘’οŽ€ο‘½οŽ₯
  • B𝑒οŽ₯
  • C√3π‘’οŽ€ο‘½οŽ₯
  • D√3𝑒οŽ₯

Q20:

Given that 𝑧=βˆ’3√3βˆ’3π‘–οŠ§, Imο€Ύπ‘§π‘§οŠ=0, and |||𝑧𝑧|||=3|𝑧|, find all the possible values of π‘§οŠ¨, expressing them in exponential form.

  • A𝑧=1√3π‘’οŠ¨οƒο‘½οŽ οŽ‘
  • B𝑧=1√3π‘’οŠ¨οƒοŽ¦ο‘½οŽ οŽ‘, 1√3π‘’ο‘½οŽ οŽ‘οƒ
  • C𝑧=1√3π‘’οŠ¨οŠ±οƒοŽ€ο‘½οŽ οŽ‘, 1√3π‘’ο‘½οŽ οŽ‘οƒ
  • D𝑧=1√3π‘’οŠ¨οŠ±οƒοŽ€ο‘½οŽ οŽ‘, 1√3π‘’ο‘½οŽ οŽ‘οƒ, 1√3π‘’οŽ¦ο‘½οŽ οŽ‘οƒ and 1√3𝑒οŽͺοŽ οŽ ο‘½οŽ οŽ‘οƒ

Q21:

Put 𝑧=3√2ο€»βˆ’πœ‹4+π‘–πœ‹4sincos in exponential form.

  • A3√2π‘’ο‘½οŽ£οƒ
  • B√22π‘’οŽ’ο‘½οŽ£οƒ
  • Cπ‘’οŽ’ο‘½οŽ£οƒ
  • D3√2π‘’οŽ’ο‘½οŽ£οƒ
  • Eβˆ’3√2π‘’οŽ’ο‘½οŽ£οƒ

Q22:

Simplify ο€»3+√3π‘–ο‡οŠ¨, giving your answer in exponential form.

  • A2√3π‘’ο‘½οŽ’οƒ
  • B4√3π‘’ο‘½οŽ’οƒ
  • C12π‘’ο‘½οŽ’οƒ
  • D12π‘’οŽ€ο‘½οŽ₯
  • E12𝑒οŽ₯

Q23:

Simplify 𝑍=βˆ’4(2+7𝑖)π‘’οŽ„οƒ, giving your answer in algebraic form.

  • Aβˆ’8+28𝑖
  • B8βˆ’28𝑖
  • C8+28𝑖
  • Dβˆ’8βˆ’28𝑖

Q24:

Put 𝑧=βˆ’7ο€»πœ‹4+π‘–πœ‹4cossin in exponential form.

  • Aβˆ’7π‘’οŽ€ο‘½οŽ£οƒ
  • Bπ‘’οŽ€ο‘½οŽ£οƒ
  • C7π‘’ο‘½οŽ£οƒ
  • D7π‘’οŽ€ο‘½οŽ£οƒ

Q25:

Given that 𝑍=10ο€Όο€Ό17πœ‹12+𝑖17πœ‹12cossin, determine the value of 1𝑍, giving your answer in exponential form.

  • A10π‘’οŽ οŽ¦ο‘½οŽ οŽ‘οƒ
  • B110π‘’οŽ¦ο‘½οŽ οŽ‘οƒ
  • C10π‘’οŽ¦ο‘½οŽ οŽ‘οƒ
  • D110π‘’οŽ οŽ¦ο‘½οŽ οŽ‘οƒ

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