# ACT Math : Complex Numbers

## Example Questions

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### Example Question #2 : How To Multiply Complex Numbers

What is the product of  and

Explanation:

Multiplying complex numbers is like multiplying binomials, you have to use foil. The only difference is, when you multiply the two terms that have  in the them you can simplify the  to negative 1. Foil is first, outside, inside, last

First

Outside:

Inside

Last

Add them all up and you get

### Example Question #3 : How To Multiply Complex Numbers

Simplify the following:

Explanation:

Begin this problem by doing a basic FOIL, treating  just like any other variable.  Thus, you know:

Recall that since .  Therefore, you can simplify further:

### Example Question #21 : Complex Numbers

Complex numbers take the form , where  is the real term in the complex number and  is the nonreal (imaginary) term in the complex number.

Distribute:

Explanation:

This equation can be solved very similarly to a binomial like . Distribution takes place into both the real and nonreal terms inside the complex number, where applicable.

### Example Question #42 : Squaring / Square Roots / Radicals

Complex numbers take the form , where  is the real term in the complex number and  is the nonreal (imaginary) term in the complex number.

Distribute and solve:

Explanation:

This problem can be solved very similarly to a binomial like .

### Example Question #43 : Squaring / Square Roots / Radicals

Complex numbers take the form , where  is the real term in the complex number and  is the nonreal (imaginary) term in the complex number.

Which of the following is equivalent to ?

Explanation:

When dealing with complex numbers, remember that .

If we square , we thus get .

Yet another exponent gives us  OR .

But when we hit , we discover that

Thus, we have a repeating pattern with powers of , with every 4 exponents repeating the pattern. This means any power of  evenly divisible by 4 will equal 1, any power of  divisible by 4 with a remainder of 1 will equal , and so on.

Thus,

Since the remainder is 3, we know that .

### Example Question #44 : Squaring / Square Roots / Radicals

Complex numbers take the form , where  is the real term in the complex number and  is the nonreal (imaginary) term in the complex number.

Simplify if possible. Leave no complex numbers in the denominator.

Explanation:

Solving this problem requires eliminating the nonreal term of the denominator. Our best bet for this is to cancel the nonreal term out by using the conjugate of the denominator.

Remember that for all binomials , there exists a conjugate  such that .

This can also be applied to complex conjugates, which will eliminate the nonreal portion entirely (since )!

Multiply both terms by the denominator's conjugate.

Simplify. Note .

FOIL the numerator.

Combine and simplify.

Simplify the fraction.

Thus, .

### Example Question #22 : Complex Numbers

Simplify the following:

Explanation:

Begin by treating this just like any normal case of FOIL. Notice that this is really the form of a difference of squares. Therefore, the distribution is very simple. Thus:

Now, recall that . Therefore,  is . Based on this, we can simplify further:

### Example Question #46 : Squaring / Square Roots / Radicals

Which of the following is equal to ?