### All ACT Math Resources

## Example Questions

### Example Question #6 : Square Of Sum

Which of the following is the square of ?

**Possible Answers:**

**Correct answer:**

Use the square of a sum pattern, substituting for and for in the pattern:

### Example Question #11 : Squaring / Square Roots / Radicals

Which real number satisfies ?

**Possible Answers:**

**Correct answer:**

Simplify the base of 9 and 27 in order to have a common base.

(3^{x})(9)=27^{2}

= (3^{x})(3^{2})=(3^{3})^{2}

=(3^{x+2})=3^{6 }

Therefore:

x+2=6

x=4

^{ }

### Example Question #2 : How To Factor A Common Factor Out Of Squares

Which of the following is a factor of ?

**Possible Answers:**

**Correct answer:**

The terms of have as their greatest common factor, so

is a prime polynomial.

Of the five choices, only is a factor.

### Example Question #12 : Squaring / Square Roots / Radicals

Simplify

**Possible Answers:**

**Correct answer:**

The easiest way to approach this problem is to break everything into exponents. is equal to and 27 is equal to . Therefore, the expression can be broken down into . When you cancel out all the terms, you get , which equals .

### Example Question #131 : Exponents

Which of the following expression is equal to

**Possible Answers:**

**Correct answer:**

When simplifying a square root, consider the factors of each of its component parts:

Combine like terms:

Remove the common factor, :

Pull the outside of the equation as :

### Example Question #2 : Squaring / Square Roots / Radicals

Which of the following is equal to the following expression?

**Possible Answers:**

**Correct answer:**

First, break down the components of the square root:

Combine like terms. Remember, when multiplying exponents, add them together:

Factor out the common factor of :

Factor the :

Combine the factored with the :

Now, you can pull out from underneath the square root sign as :

### Example Question #4 : Squaring / Square Roots / Radicals

Which of the following expressions is equal to the following expression?

**Possible Answers:**

**Correct answer:**

First, break down the component parts of the square root:

Combine like terms in a way that will let you pull some of them out from underneath the square root symbol:

Pull out the terms with even exponents and simplify:

### Example Question #4 : How To Factor A Common Factor Out Of Squares

What is,

?

**Possible Answers:**

**Correct answer:**

To find an equivalency we must rationalize the denominator.

To rationalize the denominator multiply the numerator and denominator by the denominator.

Factor out 6,

Extract perfect square 9 from the square root of 18.

### Example Question #11 : Squaring / Square Roots / Radicals

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 #2 : 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.