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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
.
Note that,
Example Question #11 : Powers And Roots Of Complex Numbers
What is the length of
?
We have
.
Hence,
.
Example Question #1 : Find The Roots Of Complex Numbers
Solve for (there may be more than one solution).
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 #4 : Find The Roots Of Complex Numbers
Recall that is just shorthand for
when dealing with complex numbers in polar form.
Express
in polar form.
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).
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
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 #1 : Find The Roots Of Complex Numbers
Determine the length of
, so
Example Question #2 : Find The Roots Of Complex Numbers
Solve for all possible solutions to the quadratic expression:
Solve for complex values of m using the aforementioned quadratic formula:
Example Question #3 : Find The Roots Of Complex Numbers
Which of the following lists all possible solutions to the quadratic expression:
Solve for complex values of using the quadratic formula:
Example Question #10 : Find The Roots Of Complex Numbers
Determine the length of .
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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