Precalculus : Powers and Roots of Complex Numbers

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Find The Roots Of Complex Numbers

Evaluate , where  is a natural number and  is the complex number .

Possible Answers:

Correct answer:

Explanation:

Note that,

 

Example Question #2 : Find The Roots Of Complex Numbers

What is the  length of 

?

Possible Answers:

Correct answer:

Explanation:

We have

.

Hence,

.

Example Question #11 : Powers And Roots Of Complex Numbers

Solve for  (there may be more than one solution).

Possible Answers:

Correct answer:

Explanation:

Solving that equation is equivalent to solving the roots of the polynomial .

Clearly, one of roots is 1.

Thus, we can factor the polynomial as 

so that we solve for the roots of .

Using the quadratic equation, we solve for roots, which are .

 

This means the solutions to  are

 

Example Question #1 : Find The Roots Of Complex Numbers

Recall that  is just shorthand for  when dealing with complex numbers in polar form. 

Express   in polar form.

Possible Answers:

Correct answer:

Explanation:

First we recognize that we are trying to solve  where .

Then we want to convert  into polar form using,

  and .

Then since De Moivre's theorem states,

  if  is an integer, we can say

.

Example Question #5 : Find The Roots Of Complex Numbers

Solve for  (there may be more than one solution).

Possible Answers:

Correct answer:

Explanation:

To solve for the roots, just set equal to zero and solve for z using the quadratic formula () :  and now setting both  and  equal to zero we end up with the answers  and  

 

Example Question #6 : Find The Roots Of Complex Numbers

Compute

 

Possible Answers:

Correct answer:

Explanation:

To solve this question, you must first derive a few values and convert the equation into exponential form: :  

 

Now plug back into the original equation and solve:   

 

Example Question #7 : Find The Roots Of Complex Numbers

Determine the length of 

Possible Answers:

Correct answer:

Explanation:

, so  

Example Question #8 : Find The Roots Of Complex Numbers

Solve for all possible solutions to the quadratic expression:

 

Possible Answers:

Correct answer:

Explanation:

Solve for complex values of m using the aforementioned quadratic formula: 

Example Question #9 : Find The Roots Of Complex Numbers

Which of the following lists all possible solutions to the quadratic expression: 

Possible Answers:

Correct answer:

Explanation:

Solve for complex values of  using the quadratic formula: 

Example Question #10 : Find The Roots Of Complex Numbers

Determine the length of .

Possible Answers:

Correct answer:

Explanation:

To begin, we must recall that . Plug this in to get . Length must be a positive value, so we'll take the absolute value: . Therefore the length is 3.

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