### All ACT Math Resources

## Example Questions

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

**Possible Answers:**

**Correct answer:**

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 #4 : How To Multiply 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 and solve:

**Possible Answers:**

**Correct answer:**

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

### Example Question #22 : 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.

Which of the following is equivalent to ?

**Possible Answers:**

**Correct answer:**

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 #3 : How To Multiply Complex Numbers

Simplify the following:

**Possible Answers:**

**Correct answer:**

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 #8 : How To Multiply Complex Numbers

Which of the following is equal to ?

**Possible Answers:**

**Correct answer:**

Remember that since , you know that is . Therefore, is or . This makes our question very easy.

is the same as or

Thus, we know that is the same as or .

### Example Question #41 : Squaring / Square Roots / Radicals

Simplify the following expression, leaving no complex numbers in the denominator.

**Possible Answers:**

**Correct answer:**

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, .