Worksheet: Arbitrary Roots of Complex Numbers

In this worksheet, we will practice using de Moivre's theorem to find the nth roots of a complex number and exploring their properties.

Q1:

Find the fourth roots of 625, giving your answers in trigonometric form.

  • A 𝑍 = 5 , 𝑍 = 5 ( ( πœ‹ ) βˆ’ 𝑖 ( πœ‹ ) ) , 𝑍 = 5 ο€Ό ο€Ό 3 πœ‹ 2  βˆ’ 𝑖 ο€Ό 3 πœ‹ 2   , 𝑍 = 5 ( ( 2 πœ‹ ) βˆ’ 𝑖 ( 2 πœ‹ ) )    οŠͺ s i n c o s s i n c o s s i n c o s
  • B 𝑍 = 2 5 , 𝑍 = 2 5 ο€» ο€» πœ‹ 2  + 𝑖 ο€» πœ‹ 2   , 𝑍 = 2 5 ο€» ο€» βˆ’ πœ‹ 2  + 𝑖 ο€» βˆ’ πœ‹ 2   , 𝑍 = 2 5 ( ( πœ‹ ) + 𝑖 ( πœ‹ ) )    οŠͺ c o s s i n c o s s i n c o s s i n
  • C 𝑍 = 2 5 , 𝑍 = 2 5 ( ( πœ‹ ) βˆ’ 𝑖 ( πœ‹ ) ) , 𝑍 = 2 5 ο€Ό ο€Ό 3 πœ‹ 2  βˆ’ 𝑖 ο€Ό 3 πœ‹ 2   , 𝑍 = 2 5 ( ( 2 πœ‹ ) βˆ’ 𝑖 ( 2 πœ‹ ) )    οŠͺ s i n c o s s i n c o s s i n c o s
  • D 𝑍 = 5 , 𝑍 = 5 ο€» ο€» πœ‹ 2  + 𝑖 ο€» πœ‹ 2   , 𝑍 = 5 ο€» ο€» βˆ’ πœ‹ 2  + 𝑖 ο€» βˆ’ πœ‹ 2   , 𝑍 = 5 ( ( πœ‹ ) + 𝑖 ( πœ‹ ) )    οŠͺ c o s s i n c o s s i n c o s s i n

Q2:

Find the fourth roots of βˆ’ 1 , giving your answers in trigonometric form.

  • A 𝑍 = ο€» ο€» πœ‹ 4  + 𝑖 ο€» πœ‹ 4    c o s s i n , 𝑍 = ο€Ό ο€Ό 3 πœ‹ 4  + 𝑖 ο€Ό 3 πœ‹ 4    c o s s i n , 𝑍 = ο€» ο€» βˆ’ πœ‹ 4  + 𝑖 ο€» βˆ’ πœ‹ 4    c o s s i n , 𝑍 = ο€Ό ο€Ό βˆ’ 3 πœ‹ 4  + 𝑖 ο€Ό βˆ’ 3 πœ‹ 4   οŠͺ c o s s i n
  • B 𝑍 = 2 ο€» ο€» πœ‹ 4  + 𝑖 ο€» πœ‹ 4    c o s s i n , 𝑍 = 2 ο€Ό ο€Ό 3 πœ‹ 4  + 𝑖 ο€Ό 3 πœ‹ 4    c o s s i n , 𝑍 = 2 ο€» ο€» βˆ’ πœ‹ 4  + 𝑖 ο€» βˆ’ πœ‹ 4    c o s s i n , 𝑍 = 2 ο€Ό ο€Ό βˆ’ 3 πœ‹ 4  + 𝑖 ο€Ό βˆ’ 3 πœ‹ 4   οŠͺ c o s s i n
  • C 𝑍 = ο€Ό ο€Ό 3 πœ‹ 4  βˆ’ 𝑖 ο€Ό 3 πœ‹ 4    s i n c o s , 𝑍 = ο€Ό ο€Ό 5 πœ‹ 4  βˆ’ 𝑖 ο€Ό 5 πœ‹ 4    s i n c o s , 𝑍 = ο€Ό ο€Ό 7 πœ‹ 4  βˆ’ 𝑖 ο€Ό 7 πœ‹ 4    s i n c o s , 𝑍 = ο€Ό ο€Ό 9 πœ‹ 4  βˆ’ 𝑖 ο€Ό 9 πœ‹ 4   οŠͺ s i n c o s
  • D 𝑍 = 2 ο€Ό ο€Ό 3 πœ‹ 4  βˆ’ 𝑖 ο€Ό 3 πœ‹ 4    s i n c o s , 𝑍 = 2 ο€Ό ο€Ό 5 πœ‹ 4  βˆ’ 𝑖 ο€Ό 5 πœ‹ 4    s i n c o s , 𝑍 = 2 ο€Ό ο€Ό 7 πœ‹ 4  βˆ’ 𝑖 ο€Ό 7 πœ‹ 4    s i n c o s , 𝑍 = 2 ο€Ό ο€Ό 9 πœ‹ 4  βˆ’ 𝑖 ο€Ό 9 πœ‹ 4   οŠͺ s i n c o s

Q3:

Determine the solution set of the equation 𝑧 = 4 ο€» √ 2 βˆ’ √ 2 𝑖   in β„‚ , expressing the solutions in exponential form.

  • A  𝑒 , 𝑒 , 𝑒       ο‘½    ο‘½    ο‘½ 
  • B  2 𝑒 , 2 𝑒 , 2 𝑒       ο‘½   ο‘½    ο‘½ 
  • C  𝑒 , 𝑒 , 𝑒      ο‘½   ο‘½   ο‘½  
  • D  2 𝑒 , 2 𝑒 , 2 𝑒       ο‘½    ο‘½    ο‘½ 

Q4:

Determine the solution set of the equation 𝑧 = 2 + 2 √ 3 𝑖  in β„‚ .

  • A  √ 3 2 + 1 2 𝑖 , βˆ’ √ 3 2 βˆ’ 1 2 𝑖 
  • B  βˆ’ √ 3 2 + 1 2 𝑖 , √ 3 2 βˆ’ 1 2 𝑖 
  • C  √ 3 βˆ’ 𝑖 , βˆ’ √ 3 + 𝑖 
  • D  √ 3 + 𝑖 , βˆ’ √ 3 βˆ’ 𝑖 

Q5:

Determine the square roots of 𝑧 , given that 𝑧 = βˆ’ 8 𝑖 .

  • A { 𝑖 , βˆ’ 𝑖 }
  • B  1 √ 2 βˆ’ 1 √ 2 𝑖 , βˆ’ 1 √ 2 + 1 √ 2 𝑖 
  • C  √ 3 2 + 1 2 𝑖 , βˆ’ √ 3 2 βˆ’ 1 2 𝑖 
  • D { 1 , βˆ’ 1 }
  • E { 2 βˆ’ 2 𝑖 , βˆ’ 2 + 2 𝑖 }

Q6:

Given that 𝑧 = βˆ’ 8 𝑖 , determine the square roots of 𝑧 without first converting it to trigonometric form.

  • A  βˆ’ 1 2 + √ 3 2 𝑖 , 1 2 βˆ’ √ 3 2 𝑖 
  • B { 𝑖 , βˆ’ 𝑖 }
  • C  1 √ 2 βˆ’ 1 √ 2 𝑖 , βˆ’ 1 √ 2 + 1 √ 2 𝑖 
  • D { 1 , βˆ’ 1 }
  • E { 2 βˆ’ 2 𝑖 , βˆ’ 2 + 2 𝑖 }

Q7:

Given that 𝑧 = βˆ’ 2 8 + 9 6 𝑖 , determine the square roots of 𝑧 without first converting it to trigonometric form.

  • A ο€Ό 2 4 2 5 βˆ’ 7 2 5 𝑖  , βˆ’ ο€Ό 2 4 2 5 βˆ’ 7 2 5 𝑖 
  • B ο€Ό βˆ’ 7 2 5 + 2 4 2 5 𝑖  , βˆ’ ο€Ό βˆ’ 7 2 5 + 2 4 2 5 𝑖 
  • C ( 8 + 6 𝑖 ) , βˆ’ ( 8 + 6 𝑖 )
  • D ο€Ό √ 2 βˆ’ 1 2 𝑖  , βˆ’ ο€Ό √ 2 βˆ’ 1 2 𝑖 
  • E ( 6 + 8 𝑖 ) , βˆ’ ( 6 + 8 𝑖 )

Q8:

Without first converting 𝑧 to trigonometric form, determine the square roots of 𝑧 , where 𝑧 = βˆ’ 8 + 2 𝑖 1 + 4 𝑖 .

  • A1, βˆ’ 1
  • B 1 βˆ’ 𝑖 , 1 + 𝑖
  • C 𝑖 , βˆ’ 𝑖
  • D 1 + 𝑖 , βˆ’ 1 βˆ’ 𝑖

Q9:

Determine the two square roots of 2 ( βˆ’ 7 βˆ’ 7 𝑖 ) βˆ’ 7 + 7 𝑖 without first converting it to trigonometric form.

  • A Β± 1
  • B Β± 𝑖
  • C Β± ο€Ώ 1 √ 2 βˆ’ √ 3 𝑖 
  • D Β± ( 1 + 𝑖 )
  • E Β± ( 1 βˆ’ 𝑖 )

Q10:

Find the square roots of βˆ’ 1 8 1 βˆ’ √ 3 𝑖 , giving your answers in trigonometric form.

  • A ( 3 0 + 𝑖 3 0 ) c o s s i n ∘ ∘ , ( 2 1 0 + 𝑖 2 1 0 ) c o s s i n ∘ ∘
  • B 9 ( 3 0 + 𝑖 3 0 ) c o s s i n ∘ ∘ , 9 ( 2 1 0 + 𝑖 2 1 0 ) c o s s i n ∘ ∘
  • C 3 ( 3 0 + 𝑖 3 0 ) c o s s i n ∘ ∘ , 3 ( 1 5 0 + 𝑖 1 5 0 ) c o s s i n ∘ ∘
  • D 3 ( 3 0 + 𝑖 3 0 ) c o s s i n ∘ ∘ , 3 ( 2 1 0 + 𝑖 2 1 0 ) c o s s i n ∘ ∘
  • E 3 ( 6 0 + 𝑖 6 0 ) c o s s i n ∘ ∘ , 3 ( 2 1 0 + 𝑖 2 1 0 ) c o s s i n ∘ ∘

Q11:

Determine, in trigonometric form, the square roots of 4 𝑒  ο‘½ οŽ₯  .

  • A 2 ο€Ό ο€Ό 5 πœ‹ 1 2  + 𝑖 ο€Ό 5 πœ‹ 1 2   c o s s i n , 2 ο€Ό ο€Ό 1 1 πœ‹ 1 2  + 𝑖 ο€Ό 1 1 πœ‹ 1 2   c o s s i n
  • B ο€Ό ο€Ό 5 πœ‹ 1 2  + 𝑖 ο€Ό 5 πœ‹ 1 2   c o s s i n , ο€Ό ο€Ό βˆ’ 7 πœ‹ 1 2  + 𝑖 ο€Ό βˆ’ 7 πœ‹ 1 2   c o s s i n
  • C ο€Ό ο€Ό 5 πœ‹ 1 2  + 𝑖 ο€Ό 5 πœ‹ 1 2   c o s s i n , ο€Ό ο€Ό 1 1 πœ‹ 1 2  + 𝑖 ο€Ό 1 1 πœ‹ 1 2   c o s s i n
  • D 2 ο€Ό ο€Ό 5 πœ‹ 1 2  + 𝑖 ο€Ό 5 πœ‹ 1 2   c o s s i n , 2 ο€Ό ο€Ό βˆ’ 7 πœ‹ 1 2  + 𝑖 ο€Ό βˆ’ 7 πœ‹ 1 2   c o s s i n
  • E 2 ο€Ό ο€Ό 7 πœ‹ 1 2  + 𝑖 ο€Ό 7 πœ‹ 1 2   c o s s i n , 2 ο€Ό ο€Ό βˆ’ 5 πœ‹ 1 2  + 𝑖 ο€Ό βˆ’ 5 πœ‹ 1 2   c o s s i n

Q12:

Use De Moivre’s theorem to find the two square roots of 1 6 ο€Ό 5 πœ‹ 3 βˆ’ 𝑖 5 πœ‹ 3  c o s s i n .

  • A Β± 4 𝑖
  • B Β± ο€» 2 √ 3 + 2 𝑖 
  • C Β± ο€Ώ 1 2 βˆ’ 1 √ 2 𝑖 
  • D Β± ο€Ώ √ 3 2 + 1 2 𝑖 

Q13:

Given that π‘₯ = 7 βˆ’ 9 𝑖 4 βˆ’ 2 𝑖 βˆ’ βˆ’ 8 βˆ’ 5 𝑖 βˆ’ 1 + 3 𝑖 , find the two square roots of π‘₯ without first expressing π‘₯ in trigonometric form.

  • A Β± ο€Ό 3 5 βˆ’ 4 5 𝑖 
  • B Β± ( 2 βˆ’ 𝑖 )
  • C Β± ο€Ό βˆ’ 4 5 + 3 5 𝑖 
  • D Β± ο€Ώ √ 3 2 βˆ’ 1 √ 2 𝑖 
  • E Β± ( βˆ’ 1 βˆ’ 2 𝑖 )

Q14:

Given that π‘₯ = 3 + 4 𝑖 , find π‘₯    .

  • A Β± ο€Ό 4 5 + 3 5 𝑖 
  • B Β± ο€½ 1 + 2 𝑖 3 
  • C Β± ο€½ 2 βˆ’ 𝑖 5 
  • D Β± ο€Ώ βˆ’ 1 √ 2 + 1 √ 2 𝑖 
  • E Β± ( 2 + 𝑖 )

Q15:

Given that π‘₯ + 𝑦 𝑖 = ο€½ 4 + 2 𝑖 1 βˆ’ 2 𝑖    , find all possible real values of π‘₯ and 𝑦 .

  • A  ο€Ό βˆ’ √ 3 , βˆ’ 1 2  , ο€Ό √ 3 , 1 2  
  • B  ο€Ό 1 4 , βˆ’ 1  , ο€Ό βˆ’ 1 4 , 1  
  • C { ( 0 , 2 ) , ( 0 , βˆ’ 2 ) }
  • D { ( 1 , 1 ) , ( βˆ’ 1 , βˆ’ 1 ) }
  • E { ( βˆ’ 3 , βˆ’ 4 ) , ( 3 , 4 ) }

Q16:

Given that 𝑧 = ο€» ο€» πœ‹ 6  + 𝑖 ο€» πœ‹ 6   s i n c o s , determine the cubic roots of ο€Ή 𝑧   .

  • A ο€Ό ο€Ό 7 πœ‹ 9  + 𝑖 ο€Ό 7 πœ‹ 9   c o s s i n , ο€Ό ο€Ό βˆ’ 8 πœ‹ 9  + 𝑖 ο€Ό βˆ’ 8 πœ‹ 9   c o s s i n , ο€Ό ο€Ό βˆ’ 5 πœ‹ 9  + 𝑖 ο€Ό βˆ’ 5 πœ‹ 9   c o s s i n
  • B ο€Ό ο€Ό 7 πœ‹ 9  + 𝑖 ο€Ό 7 πœ‹ 9   c o s s i n , ο€Ό ο€Ό βˆ’ 5 πœ‹ 9  + 𝑖 ο€Ό βˆ’ 5 πœ‹ 9   c o s s i n , ο€» ο€» πœ‹ 9  + 𝑖 ο€» πœ‹ 9   c o s s i n
  • C ο€Ό ο€Ό 5 πœ‹ 1 8  + 𝑖 ο€Ό 5 πœ‹ 1 8   c o s s i n , ο€Ό ο€Ό 1 7 πœ‹ 1 8  + 𝑖 ο€Ό 1 7 πœ‹ 1 8   c o s s i n , ο€Ό ο€Ό βˆ’ 7 πœ‹ 1 8  + 𝑖 ο€Ό βˆ’ 7 πœ‹ 1 8   c o s s i n
  • D ο€Ό ο€Ό 5 πœ‹ 9  + 𝑖 ο€Ό 5 πœ‹ 9   c o s s i n , ο€Ό ο€Ό βˆ’ 7 πœ‹ 9  + 𝑖 ο€Ό βˆ’ 7 πœ‹ 9   c o s s i n , ο€» ο€» βˆ’ πœ‹ 9  + 𝑖 ο€» βˆ’ πœ‹ 9   c o s s i n
  • E ο€Ό ο€Ό βˆ’ 7 πœ‹ 9  + 𝑖 ο€Ό βˆ’ 7 πœ‹ 9   c o s s i n , ο€Ό ο€Ό βˆ’ 4 πœ‹ 9  + 𝑖 ο€Ό βˆ’ 4 πœ‹ 9   c o s s i n , ο€» ο€» βˆ’ πœ‹ 9  + 𝑖 ο€» βˆ’ πœ‹ 9   c o s s i n

Q17:

Given that 𝑧 = βˆ’ 5 + 4 𝑖 βˆ’ 5 𝑖 2 + 7 𝑖 βˆ’ 2 𝑖 + 7 𝑖    , find the square roots of 𝑧 in trigonometric form.

  • A ο€» ο€» πœ‹ 6  + 𝑖 ο€» πœ‹ 6   c o s s i n , ο€» ο€» βˆ’ πœ‹ 6  + 𝑖 ο€» βˆ’ πœ‹ 6   c o s s i n
  • B ο€» ο€» πœ‹ 4  + 𝑖 ο€» πœ‹ 4   c o s s i n , ο€Ό ο€Ό βˆ’ 3 πœ‹ 4  + 𝑖 ο€Ό βˆ’ 3 πœ‹ 4   c o s s i n
  • C ο€» ο€» βˆ’ πœ‹ 4  + 𝑖 ο€» βˆ’ πœ‹ 4   c o s s i n , ο€Ό ο€Ό 3 πœ‹ 4  + 𝑖 ο€Ό 3 πœ‹ 4   c o s s i n
  • D ο€» ο€» πœ‹ 3  + 𝑖 ο€» πœ‹ 3   c o s s i n , ο€» ο€» βˆ’ πœ‹ 3  + 𝑖 ο€» βˆ’ πœ‹ 3   c o s s i n
  • E ο€» ο€» πœ‹ 2  + 𝑖 ο€» πœ‹ 2   c o s s i n , ο€» ο€» βˆ’ πœ‹ 2  + 𝑖 ο€» βˆ’ πœ‹ 2   c o s s i n

Q18:

Find the two square roots of π‘₯ = 8 + 𝑖 βˆ’ 1 βˆ’ 2 𝑖 + βˆ’ 9 + 2 𝑖 βˆ’ 4 βˆ’ 𝑖 in β„‚ , without first expressing π‘₯ in trigonometric form.

  • A Β± ( 1 + 𝑖 )
  • B Β± ( 1 βˆ’ 𝑖 )
  • C Β± 1
  • D Β± 𝑖

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