### All ACT Math Resources

## Example Questions

### Example Question #1 : Complex Numbers

Subtract from , given:

**Possible Answers:**

**Correct answer:**

A complex number is a combination of a real and imaginary number. To subtract complex numbers, subtract each element separately.

In equation , is the real component and is the imaginary component (designated by ). In equation , is the real component and is the imaginary component. Solving for ,

### Example Question #1 : Complex Numbers

Simplify the exponent,

.

**Possible Answers:**

**Correct answer:**

When you have an exponent on the outside of parentheses while another is on the inside of the parentheses, such as in , multiply the exponents together to get the answer: .

This is different than when you have two numbers with the same base multiplied together, such as in . In that case, you add the exponents together.

### Example Question #3 : 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:**

Solving this equation is very similar to solving a linear binomial like . To solve, just combine like terms, being careful to watch for double negatives.

### Example Question #4 : 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:**

A problem like this can be solved similarly to a linear binomial like /

### Example Question #5 : 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 equations simplifies into ?

**Possible Answers:**

**Correct answer:**

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

### Example Question #6 : Complex Numbers

Suppose and

Evaluate the following expression:

**Possible Answers:**

**Correct answer:**

Substituting for and , we have

This simplifies to

which equals

### Example Question #1 : Complex Numbers

What is the solution of the following equation?

**Possible Answers:**

**Correct answer:**

A complex number is a combination of a real and imaginary number. To add complex numbers, add each element separately.

First, distribute:

Then, group the real and imaginary components:

Solve to get:

### Example Question #8 : Complex Numbers

What is the sum of and given

and

?

**Possible Answers:**

**Correct answer:**

A complex number is a combination of a real and imaginary number. To add complex numbers, add each element separately.

In equation , is the real component and is the imaginary component (designated by ).

In equation , is the real component and is the imaginary component.

When added,

### Example Question #9 : Complex Numbers

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

Simplify:

**Possible Answers:**

**Correct answer:**

When adding or subtracting complex numbers, the real terms are additive/subtractive, and so are the nonreal terms.

### Example Question #10 : Complex Numbers

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

Can you add the following two numbers: ? If so, what is their sum?

**Possible Answers:**

**Correct answer:**

Complex numbers take the form *a + bi*, where *a* is the real term in the complex number and *bi* is the nonreal (imaginary) term in the complex number. Taking this, we can see that for the real number 8, we can rewrite the number as , where represents the (zero-sum) non-real portion of the complex number.

Thus, any real number can be added to any complex number simply by considering the nonreal portion of the number to be .