# Precalculus : Powers and Roots of Complex Numbers

## Example Questions

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### Example Question #1 : Powers And Roots Of Complex Numbers

Find the magnitude of the complex number described by .

Explanation:

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

,

where our complex number is in the form .

Therefore,

### Example Question #1 : Powers And Roots Of Complex Numbers

Find the magnitude of :

, where the complex number satisfies .

Explanation:

Note for any complex number z, we have:

.

Let .  Hence

Therefore:

This gives the result.

### Example Question #3 : Powers And Roots Of Complex Numbers

What is the magnitude of ?

Explanation:

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

, where .

Therefore we get,

.

Now to find

.

### Example Question #4 : Powers And Roots Of Complex Numbers

Simplify

Explanation:

We can use DeMoivre's formula which states:

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

### Example Question #5 : Powers And Roots Of Complex Numbers

Explanation:

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

### Example Question #6 : Powers And Roots Of Complex Numbers

Evaluate

Explanation:

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

### Example Question #7 : Powers And Roots Of Complex Numbers

Evaluate

Explanation:

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

### Example Question #8 : Powers And Roots Of Complex Numbers

Use DeMoivre's Theorem to evaluate the expression .

Explanation:

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.

### Example Question #9 : Powers And Roots Of Complex Numbers

Evaluate:

Explanation:

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

### Example Question #10 : Powers And Roots Of Complex Numbers

Explanation:

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:

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