Evaluate Powers of Complex Numbers Using DeMoivre's Theorem - 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

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

,

where our complex number is in the form .

Therefore,

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