### All ACT Math Resources

## Example Questions

### Example Question #1 : How To Add 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 incorrect?

**Possible Answers:**

**Correct answer:**

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

Thus, to balance the equation, add like terms on the left side.

### Example Question #1 : How To Divide Complex Numbers

Simplify:

**Possible Answers:**

**Correct answer:**

Multiply both numberator and denominator by :

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

Evaluate:

**Possible Answers:**

**Correct answer:**

First, divide 100 by as follows:

Now dvide this result by :

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

Evaluate:

**Possible Answers:**

**Correct answer:**

First, divide 100 by as follows:

Now, divide this by :

### Example Question #31 : Squaring / Square Roots / Radicals

Evaluate:

**Possible Answers:**

**Correct answer:**

First, evaluate :

Now divide this into :

### Example Question #5 : How To Divide Complex Numbers

Evaluate:

**Possible Answers:**

**Correct answer:**

First, evaluate using the square pattern:

Divide this into :

### Example Question #2 : How To Divide 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.

Simplify:

**Possible Answers:**

**Correct answer:**

This problem can be solved very similarly to a binomial such as . In this case, both the real and nonreal terms in the complex number are eligible to be divided by the real divisor.

, so

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

Simplify by using conjugates:

**Possible Answers:**

**Correct answer:**

Solving this problem using a conjugate is just like conjugating a binomial to simplify a denominator.

Multiply both terms by the denominator's conjugate.

** **Simplify. Note** .**

** **Combine and simplify.

Simplify the numerator.

** **The prime denominator prevents further simplifying.

Thus, .

### Example Question #11 : Complex Numbers

Simplify:

**Possible Answers:**

**Correct answer:**

This problem can be solved in a way similar to other kinds of division problems (with binomials, for example). We need to get the imaginary number out of the denominator, so we will multiply the denominator by its conjugate and multiply the top by it as well to preserve the number's value.

Then, recall by definition, so we can simplify this further:

This is as far as we can simplify, so it is our final answer.

### Example Question #1 : How To Multiply Complex Numbers

The solution of is the set of all real numbers such that:

**Possible Answers:**

**Correct answer:**

Square both sides of the equation:

Then Solve for x:

Therefore,

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