Powers and Roots of Complex Numbers - Pre-Calculus

Note for any complex number z, we have:

.

Let . Hence

Therefore:

This gives the result.

To find the magnitude of a complex number we use the following formula:

, where .

Therefore we get,

.

Now to find

.

We can use DeMoivre's formula which states:

Now plugging in our values of and we get the desired result.

First, convert this complex number to polar form.

Since the point has a positive real part and a negative imaginary part, it is located in quadrant IV, so the angle is .

This gives us

To evaluate, use DeMoivre's Theorem:

DeMoivre's Theorem is

We apply it to our situation to get.

simplifying

, is coterminal with since it is an even multiple of

First convert this complex number to polar form:

so

Since this number has positive real and imaginary parts, it is in quadrant I, so the angle is

So we are evaluating

Using DeMoivre's Theorem:

DeMoivre's Theorem is

We apply it to our situation to get.

First convert this point to polar form:

Since this number has a negative imaginary part and a positive real part, it is in quadrant IV, so the angle is

We are evaluating

Using DeMoivre's Theorem:

DeMoivre's Theorem is

We apply it to our situation to get.

which is coterminal with since it is an odd multiplie

First, convert this complex number to polar form:

Since the real part is positive and the imaginary part is negative, this is in quadrant IV, so the angle is

So we are evaluating

Using DeMoivre's Theorem:

DeMoivre's Theorem is

We apply it to our situation to get.

is coterminal with since it is an even multiple of

First, convert the complex number to polar form:

Since both the real and the imaginary parts are positive, the angle is in quadrant I, so it is

This means we're evaluating

Using DeMoivre's Theorem:

DeMoivre's Theorem is

We apply it to our situation to get.

First, evaluate . We can split this into which is equivalent to

\[We can re-write the middle exponent since is equivalent to \]

This comes to

Evaluating sine and cosine at is equivalent to evaluating them at since

This means our expression can be written as:

First convert the complex number into polar form:

Since the real part is negative but the imaginary part is positive, the angle should be in quadrant II, so it is

We are evaluating

Using DeMoivre's Theorem:

DeMoivre's Theorem is

We apply it to our situation to get.

simplify and take the exponent

is coterminal with since it is an odd multiple of pi

Note that,

We have

.

Hence,

.

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

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

.

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

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:

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

, so

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

Solve for complex values of using the quadratic formula: