# CUNY Assessment Tests (CATs)

## So, you’re preparing to take the CUNY Assessment Tests?

Awesome!

You’ve found the right page. We will answer every question you have and tell you exactly what you need to study to pass the CATs.

## Quick Facts

Get the “need to know” information at a quick glance.

**Overview**

The CATs were developed by the City University of New York to evaluate students who are returning to college or entering college for the first time.

Your results on the CATs can be used to help four-year colleges decide whether you have the aptitude to succeed in their classes. The CATs can also be used to determine whether it will be necessary for you to complete some non-credit remedial coursework in reading, writing, mathematics, or English as a second language before entering a degree program.

Studying for the CATs can help students to avoid taking any unnecessary remedial tests. This, in turn, can save them time and money!

**Format**

The CAT in Reading Test (CATR) is a computer-based multiple-choice test. You will answer 20 multiple choice questions on this test. Luckily, there is no time limit on the Reading test!

On the CUNY Writing Test (CATW), you will be given a 250-300 word passage, to which you will respond by writing an essay. You will have 90 minutes to complete the CATW.

The CUNY Mathematics Test (CATM) is broken into two parts: the Elementary Algebra exam, which contains 12 questions, and the College Level Math exam, which contains 20 questions. Like the CATR, the CATM is also untimed.

**Cost**

Students do not take the CATs until they are accepted into a school. You do not have to pay a fee to take the exam for your school.

**Scoring**

The following table shows the passing scores for the CATs.

On the CATs, you only need to worry about making a passing score. Scoring an 80 as opposed to a 65 really makes no difference. Each one of your questions must be answered and scored. You will not have the ability to skip a question without answering it.

*You neither pass nor fail the College Level Math Test. The test is used by schools to determine whether you are ready to take more advanced math classes. If you are interested in taking advanced math classes, you should contact the Admissions department at your school for more information about how your particular school uses scores on the College Level Math Test.

**Study Time**

Different students will need to spend different amounts of time studying for the CATs. You can use our questions and materials to evaluate which areas you are weakest in. If you feel weak in several areas and need to brush up on many concepts, you should definitely plan to give yourself lots of time to study.

All students should focus on their areas of weakness as they study and you should never begin cramming for the exam a few days before.

As you study, take a quick break every 15-20 minutes. Stretch, drink some water, and let your mind rest for a couple of minutes before resuming your study session.

To keep yourself on track, schedule a time to study each day. Don’t try to focus on too many concepts in one study session. It’s better to understand a concept thoroughly than to “sort of” be familiar with several concepts. In the long run, you will not answer as many questions correctly if you do not have a full grasp of the concepts.

Try to make studying as comfortable and stress-free as possible. Study in a place where you can focus. Wear comfy sweatpants! Some students need silence to study, and others prefer soft music in the background. Studying with a friend can also help you keep your commitment to studying.

**What test takers wish they would’ve known:**

- Get plenty of rest the night before the exam.
- Prepare to arrive at the testing center 15 minutes early. You do not want to feel rushed and panicked before the exam.
- Wear comfortable layers of clothing. It’s a good idea to bring a sweater in case the testing center is cold.
- Check your testing location and time in advance. It’s also a good idea to make sure that you know how to get to the testing center several days before your test date.
- Be sure to bring your ID, two #2 pencils, and a (non-electronic) dictionary. You may use the dictionary on the Writing portion of the exam.

## Reading Test

You will have an unlimited amount of time to answer the 20 multiple-choice questions on the Reading test.

The Reading test focuses on the following concepts:

- Structure
- Main Idea and Theme
- Supporting Details
- Point of View
- Vocabulary

So, let’s talk about them.

**Structure**

When taking the CUNY Reading test, you may encounter a variety of different text structures. Text structure refers to the way that the information in a passage, or other text, is organized. Identifying the structure of a passage can help you to understand the ideas in the passage and how they relate to each other.

Let’s look at some examples of different text structures.

**Descriptive Structure:**In a descriptive text, the author describes an idea in detail. An encyclopedia article is an example of a text with a descriptive structure.

**Sequential Structure:**The author of this type of text gives a sequence of events in chronological order. Most fiction stories and novels have a sequential structure.

**Compare and Contrast:**The author compares and/or contrasts two or more ideas. An author compares ideas to highlight how they are alike and contrasts ideas to show how they are different. A passage explaining the difference between bottled water and tap water would have a compare and contrast structure.

**Cause and Effect:**In this type of text, the author shows how events or conditions bring about other events or conditions. A passage about how hurricanes can create tornadoes would probably have this type of structure.

**Problem and Solution:**The author of the passage introduces a problem or asks a question and gives answers or solutions. An example of this type of text would be an article describing how to keep wild rabbits from entering a garden and eating the vegetables.

**Main Idea and Theme**

As you read passages on the Reading test, it will be important to identify the main ideas and themes of passages. Let’s review these concepts.

**Finding the Main Idea**

The main idea can be described as what the text is mostly about. Main idea questions may contain phrases such as “primarily about,” “best summary,” and “focuses on the topic of.”

Take a look at the following passage and try to answer the question on your own:

*Chinese painters have always avoided mixing colors so far as possible. From malachite, they obtained several shades of green, from cinnabar or sulphide of mercury, a number of reds. They knew also how to combine mercury, sulfur, and potash to produce vermilion. From peroxide of mercury they drew coloring powders which furnished shades ranging from brick red to orange-yellow. During the T’ang dynasty coral was ground to secure a special red, while white was extracted from burnt oyster shells. White lead was later substituted for this lime white. Carmine lake they obtained from madder, yellows from the sap of the rattan, blues from indigo. To these must be added the different shades of Chinese ink and lastly, gold in leaf and in powder.*

*The passage is primarily about:*

*how to make natural paints.**the ideas which inspired early Chinese painters to create artwork.**why colors should not be mixed on a painting.**the materials Chinese painters have used to make paint.*

Choice A is incorrect because this text is not meant to teach you a skill. Although Chinese painters are mentioned in the text, the author does not mention anything about inspiration, so B is incorrect. The passage mentions that the painters being avoided mixing colors, but it does not provide information about why colors should not be mixed; C is incorrect.

Choice D correctly describes the main idea.

**Identifying a Theme**

The theme of a text is the overall message that the author is trying to relay. The theme is typically not presented outright the same way the main idea is. To find the theme, ask yourself if the author is trying to teach you a lesson or make an overall statement about life.

Here are some examples of common themes:

- Good versus evil
- Change versus tradition
- Coming of age
- Loss of innocence
- The will to survive
- The importance of courage

Read the following passage and think about its theme:

*A Lion was awakened from sleep by a Mouse running over his face. Rising up in anger, he caught him and was about to kill him, when the Mouse piteously entreated, saying: “If you would only spare my life, I would be sure to repay your kindness.” The Lion laughed and let him go. It happened shortly after this that the Lion was caught by some hunters, who bound him by strong ropes to the ground. The Mouse, recognizing his roar, came up and gnawed the rope with his teeth, and, setting him free, exclaimed: “You ridiculed the idea of my ever being able to help you, not expecting to receive from me any repayment of your favor; but now you know that it is possible for even a Mouse to confer benefits on a Lion.”*

There are several possible themes for this passage:

- No one is too insignificant to help
- The rewards of mercy
- Do not judge someone by his or her appearance
- Help can come from unexpected sources
- The benefits of being kind to others

These are all concepts that the passage teaches you, even though they are not stated outright.

**Supporting Details**

The passages on the Reading test will include details to support their themes and main ideas. You will need to find details directly in the text to answer questions. It is also important that you understand why the details are there and how they relate to other ideas in the passages.

Details may be included to help you imagine a scene or understand a character. They may also be included as evidence to help the author support his or her opinion or point.

Think back to the fable you just read about the lion and the mouse. “The Lion laughed and let him go” is included in the passage. Why?

The author includes this little detail about the lion laughing to support the idea that the lion thinks it is very unlikely that the mouse could ever help him. Therefore, the detail supports the idea that help can come from unexpected sources or those viewed as insignificant.

**Point of View**

On the Reading test, you will be asked questions that require you to think about the point of view. Let’s review this concept.

**Identifying the Perspective**

Understanding the perspective of a passage can help you to understand why the author wrote the passage, as well as the ideas or events in the passage.

Take a look at the following table, which shows different perspectives, or viewpoints:

**Identifying the Author’s Purpose**

Authors have different purposes for writing texts. On the Reading test, you should be able to identify the author’s purpose when you encounter a passage.

Here are the most common purposes of writing:

- To inform
- To persuade
- To entertain

Take a look at the following text:

“People really should be more concerned about the trees in this community. Trees are truly magnificent. They provide shade, oxygen, homes for birds, and natural beauty. If we stand up together, we can convince local authorities to change the laws about how many trees homeowners are allowed to cut down.”

The author’s purpose is to persuade. He or she wants to persuade the reader that the trees should not be cut down and that the laws about cutting down trees should be changed.

**Identifying Tone and Attitude**

The author’s attitude is the feeling he or she has about a subject. The author’s tone is relayed through word choice and helps the reader to understand his or her attitude.

Consider this sentence from the previous paragraph: *“Trees are truly magnificent.”*

This sentence shows that the author’s attitude towards trees is very positive. The word “magnificent” shows that the author uses a praising tone when speaking about the trees.

Review the following chart which includes some tone words as well as some example text excerpts. Pay attention to the underlined words in the example text excerpts; these words help develop the tone.

**Vocabulary**

As you can see, the author’s word choice really helps to develop a text. But what if you don’t understand some of the words in a passage? Let’s look at some strategies for determining the meanings of unknown words.

**Context Clues**

Context clues are hints within a text that help readers to determine the meanings of terms and phrases.

Consider the following sentences. Although you may not know the meaning of the words in bold text, the underlined text provides context clues for those words.

**Pipefish**have tubular snouts and are similar in appearance to__seahorses__, which are members of the same subfamily.- His speech seemed rather
**pompous**, but he should really have shown__humility__and__modesty__instead. - Her
**ebullient**attitude made those around her feel__cheerful__as well.

While reading the CATs passages, be sure to look for context clues.

**Using Word Parts to Determine Meaning**

If you are having trouble finding context clues, you can also look at the parts of a word to determine its meaning.

Review the following table which includes some common prefixes, which are added to the beginning of a word, and suffixes, which are added to the end of the word. The table also includes some root words.

Knowing these word parts can help you to break an unknown word apart in order to analyze and understand it.

And that’s some basic info about the Reading test.

## Writing Test

The CAT Writing will include one passage. You will have 90 minutes to read the passage and prepare a written response. The directions will ask you to include a summary of the passage in your response and to build upon at least one of the ideas in the passage.

The Writing test focuses on the following concepts:

- Summarization
- Developing Paragraphs
- Structure and Flow
- Effective Word Choice
- Grammar
- Revision

So, let’s talk about them.

**Summarization**

Summarization is an extremely important skill to demonstrate on the Writing CAT for two reasons; you will need to summarize a passage in writing and you will need to be able to summarize the passage in your mind in order to understand what it is mainly about. Only then can you write an appropriate response.

Let’s look at an example passage. As you read, create a summary of the passage in your mind by deciding what the *most important *ideas are.

*Involuntary attention there is a conflict either between the will and interest or between the will and the mental inertia or laziness, which has to be overcome before we can think with any degree of concentration. Interest says, “Follow this line, which is easy and attractive, or which requires but little effort—follow the line of least resistance.” Will says, “Quit that line of dalliance and ease, and take this harder way which I direct—cease the line of least resistance and take the one of greatest resistance.” When daydreams and “castles in Spain” attempt to lure you from your lessons, refuse to follow; shut out these vagabond thoughts and stick to your task. When intellectual inertia deadens your thought and clogs your mental stream, throw it off and court forceful effort. If wrong or impure thoughts seek entrance to your mind, close and lock your mental doors to them. If thoughts of desire try to drive out thoughts of duty, be heroic and insist that thoughts of duty shall have right of way. In short, see that you are the master of your thinking, and do not let it always be directed without your consent by influences outside of yourself.*

*It is just at this point that the strong will wins victory and the weak will breaks down. Between the ability to control one’s thoughts and the inability to control them lies all the difference between right actions and wrong actions; between withstanding temptation and yielding to it; between an inefficient purposeless life and a life of purpose and endeavor; between success and failure. For we act in accordance with those things which our thought rests upon. Suppose two lines of thought represented by A and B, respectively, lie before you; that A leads to a course of action difficult or unpleasant, but necessary to success or duty, and that B leads to a course of action easy or pleasant, but fatal to success or duty. Which course will you follow—the rugged path of duty or the easier one of pleasure? The answer depends almost wholly, if not entirely, on your power of attention. If your will is strong enough to pull your thoughts away from the fatal but attractive B and hold them resolutely on the less attractive A, then A will dictate your course of action, and you will respond to the call for endeavor, self-denial, and duty; but if your thoughts break away from the domination of your will and allow the beckoning of your interests alone, then B will dictate your course of action, and you will follow the leading of ease and pleasure. For our actions are finally and irrevocably dictated by the things we think about.*

In order to create a summary of a passage, such as the one above, it may be helpful to write down a brief note whenever you encounter a key idea.

For the given passage, your notes might look something like this:

- Voluntary attention – internal conflict
- Will vs. laziness/being disinterested
- Important to focus
- Two choices – pay attention or let mind wander

When summarizing the passage you may paraphrase ideas in your own words and/or use direct quotes from the passage. Be sure to use quotation marks around direct quotes.

The following text is an example of a brief summary of the passage. Notice that it sticks to the key ideas.

*The passage describes the concept of voluntary attention, which is the type of attention required to focus on a task that is difficult to focus on. According to the passage, focusing on a task can be strenuous because the task is disinteresting or because one must fight “mental inertia or laziness.” However, one has only two choices when facing such a task; to give in and let the mind wander, or to use one’s willpower to concentrate on the task.*

**Developing Paragraphs**

After you read the passage and are able to summarize it, it’s time to start thinking about the paragraphs you will include in your response.

**Planning Paragraphs**

Instead of diving straight into writing your response, take a few minutes to plan out your paragraphs. This is called prewriting. Brainstorm your ideas and order them logically.

Here is an example of how you might plan your paragraphs:

*Introduction – clearly state the point of view on the topic and summarize what the author says about the topic.*

*Second paragraph – use evidence from the text and agree or disagree with how it supports the author’s key points and main idea. Use evidence from your own life to agree or disagree with the main idea and key points.*

*Third paragraph – use evidence from the text and agree or disagree with how it supports a different point in the passage. Use evidence from your own life to agree or disagree with the point.*

*Fourth paragraph – use evidence from the text and agree or disagree with how it supports a third point in the passage. Use evidence from your own life to agree or disagree with the point.*

*Conclusion – summarize your point of view again in different words and comment on whether or not the overall evidence in the passage supports the author’s key points and main idea.*

Now that you know how to plan out your response, let’s look at the elements of a well-written introductory paragraph.

**Introductory Paragraphs**

An appropriate introductory paragraph should:

- Respond directly to the author’s ideas
- Include a statement expressing a clear opinion about the author’s ideas
- Summarize the author’s ideas

An introductory paragraph should NOT:

- Stray from the author’s main ideas
- Lack a thesis statement you can build on
- Present your opinion(s) in a vague way that the reader cannot understand

**Elements of Good Response Paragraphs**

Each response paragraph should:

- Communicate an opinion or idea that you have about the text
- Refer to information in the text
- Support your point of view with evidence from your own experience and education

Remember, you are aiming to balance the content of your paragraphs so that both your ideas and the author’s ideas are stated and supported.

**Conclusion Paragraphs**

When writing your conclusion paragraph, reiterate your main point without stating it exactly as you did in your introduction. Remember to summarize the evidence that you use to support your point and leave the reader with a strong message to consider. As with all paragraphs, be sure to refer directly to the passage.

A conclusion paragraph should not introduce new ideas; new ideas should only be introduced in the introduction paragraph and the supporting paragraphs. Conclusions paragraphs should wrap up ideas neatly instead of leaving the reader feeling that you have not been able to express all of your ideas.

**Structure and Flow**

On the Writing test, you should make sure that your sentences and paragraphs are structured appropriately. A strong writer is able to write sentences and paragraphs which “flow” or relate well to one another. Strong writers also avoid choppy and run-on sentences.

**Sentence Structure**

Let’s look at some specific concepts related to sentence structure and how you can apply them on the test:

**Run-on sentences should be broken into multiple sentences.**- Example:
*The author of the passage has clearly spent a lot of time contemplating the key idea that willpower is often a necessary element of concentration and he makes it clear that a person who is applying voluntary attention to a task has chosen to use his will in order to concentrate on the task at hand instead of allowing his mind to stray.* - Correction:
*The author of the passage has clearly spent a lot of time contemplating the key idea that willpower is often a necessary element of concentration. This is supported by his idea that a person who is applying voluntary attention to a task has chosen to use his will in order to concentrate on the task at hand instead of allowing his mind to stray.*

- Example:
**Choppy or incomplete sentences should be combined to express complete thoughts.**- Example:
*Because the author explains that a person has to choose between two lines of thought. And therefore, the reader can better understand the author’s point that concentration is a matter of will.* - Correction:
*The author illustrates that concentration is a matter of will by showing that a person has to choose between two lines of thought.*

- Example:
**Sentences should use connecting words and phrases such as “furthermore,” “therefore,” “as a result,” and “additionally.”**- Example:
*The author illustrates that concentration is a matter of will by showing that a person has to choose between two lines of thought. I agree with the point that he makes about having to choose between two lines of thought.* - Correction:
*The author illustrates that concentration is a matter of will by showing that a person has to choose between two lines of thought. Because I have often felt my attention being pulled in two opposite directions at once, I identify with this point.*

- Example:

**Connecting Paragraphs**

Just as sentences should be connected with appropriate words and phrases, paragraphs should be connected as well. It is disorienting to the reader if you move from one idea to another without relating the two ideas.

Consider the following text:

*The author illustrates that concentration is a matter of will by showing that a person has to choose between two lines of thought. Because I have often felt my attention being pulled in two opposite directions at once, I identify with this point. An example of a time when I experienced this phenomenon was the day that I was sitting in my American History class. That day, the line of thought that the author refers to as “B” kept drifting to ideas about what I was going to eat for dinner and whether my friend would call me later that afternoon. At the same time, the line of thought that the author labels “A” was filled with boring ideas about a civil war. While I knew that I should stick to line “A,” I found line “B” to be much more attractive and interesting.*

*The author says that laziness can also be a problem when a person needs to concentrate on a task. It is true that a slothful mind can make paying attention to a task more difficult.*

Here, the writer expressed an idea in the first paragraph and began a second paragraph without connecting it to the first paragraph. Consider how the following text would make a better introduction to the second paragraph because it relates to the first paragraph.

*Clearly, the mind is torn between line “A” and line “B” when a task is disinteresting. However, the author states that “laziness,” as well as disinterest, can also present a problem for someone who needs to concentrate.*

**Effective Word Choice**

Now, let’s take a closer look at the words that make up paragraphs. When writing your response on the test, it is important to choose words that are precise and formal. You should also vary the words that you choose to include in your essay.

**Precision**

It’s important to avoid vague language in your writing. Words that are imprecise make an essay more difficult for the reader to understand. Consider the following sentences:

- I was experiencing distracting thoughts and I
__thought about how__I had left my keys in my car__when I was not really thinking about it__. - I was experiencing distracting thoughts and I
__suddenly remembered that__I had__absent-mindedly__left my keys in my car.

Notice the underlined text and how it is more precise in the second sentence. In the second sentence, it is much easier to understand what the writer is relaying.

**Variety**

Take a look at sentence one again:

- I was experiencing distracting
__thoughts__and I__thought__about how I had left my keys in my car when I was not really__thinking__about it.

The writer keeps repeating the verb *think* (although in a couple of different tenses). How boring!

When you write, try to use synonyms for a word instead of repeating the same term multiple times. By using synonyms, you can make your writing more interesting and show off your vocabulary skills.

Here are a few examples of synonyms. Consider how you could use these synonyms in your writing:

- Think – wonder, contemplate, remember, muse
- Great – excellent, superb, fantastic, lovely
- Walked – wandered, meandered, moseyed
- Ran – sprinted, scurried, scampered, sped
- Storm – downpour, rainstorm, deluge, thunderstorm

**Formal Language**

Any time you write for a school-related task or test, including the Writing test, you should use formal language. Take a look at how informal language, which is not appropriate, can be improved:

**Grammar and Punctuation**

Next, we’ll take a look at grammar and punctuation. You will need to use correct grammar and punctuation when you take the Writing CAT.

**Pronoun-Antecedent Agreement**

An **antecedent** is the noun to which a **pronoun **(he, she, it, they) refers to. Pronouns and their antecedents must agree in number and gender.

Look at the examples below. The antecedents are in bold text and the pronouns are italicized.

**Incorrect: **Like the other students, **Cody** says that *they* would like to attend the seminar.

**Correct: **Like the other students, **Cody** says that *he* would like to attend the seminar.

**Incorrect: **Although the black puppy is my favorite, none of the **puppies** are aggressive, and *it* may be safely handled.

**Correct: **Although the black puppy is my favorite, none of the **puppies** are aggressive, and *they *may be safely handled.

**Subject-Verb Agreement**

When the **subject** of a sentence is plural, the **verb** describing its action should also be plural. If a subject is singular, the verb which describes it should also be singular.

Take a look at the following examples. In each example, the subject is in bold text and the verb is underlined.

**Incorrect:** **Creativity**, particularly with regard to subject matter and word choice, *are* needed in order to score well on the poetry assignment.

**Correct:** **Creativity**, particularly with regard to subject matter and word choice, *is*

needed in order to score well on the poetry assignment.

**Incorrect: **Each **one** of the travelers *have* chosen a room.

**Correct: **Each **one** of the travelers *has* chosen a room.

**Commas**

Commas are used for different reasons and you should use them correctly on the Writing CAT. We’ll review those reasons as well as some examples of when to use commas.

**Colons, Semicolons, and End Marks**

There are three different end marks used to punctuate sentences: periods, exclamation points, and question marks. Review them below:

**Period (.)**: Used to declare an idea or give a command

Examples:

- Please come home soon.
- Based on the article, the treatment can help patients.

**Exclamation point (!)**: Used to show urgency or a strong emotion, such as fear, surprise, or excitement

Examples:

- I hope he will hurry!
- You never pay attention!

**Question mark (?)**: Used to make an inquiry

Examples:

- Will you attend the reading at the bookstore?
- Which painting is your favorite?

**Revision**

When writing your essay, be sure to leave yourself about 10 minutes to revise your work. Although the majority of your time on the Writing CAT should be spent writing your essay, it’s important to go back and look for errors in spelling, grammar, and punctuation.

While you check over your work, you should also make sure that your sentences and paragraphs are well-connected and that your word choice is appropriate and varied.

*Do not* reorganize or rewrite large sections of your essay. If you choose to do this, you may run out of time and end up submitting an incomplete essay. Remember, this section of the test is timed.

And that’s some basic info about the Writing test.

## Mathematics Test: Elementary Algebra

The Elementary Algebra test has 12 multiple-choice questions. You will have an unlimited amount of time to complete this test.

It will be very important to be familiar with the following concepts as you take the test:

- Order of Operations
- Monomials
- Polynomials
- Roots and Exponents
- Equations and Inequalities

So, let’s talk about them.

**Order of Operations**

When solving problems on the Elementary Algebra test, it is important to perform operations, such as subtraction and multiplication, in the correct order.

You may have used the acronym “PEMDAS” while solving problems before.

The acronym stands for *P**arentheses, **E**xponents, **M**ultiplication/**D**ivision, **A**ddition/**S**ubtraction*. “Please Excuse My Dear Aunt Sally” is a phrase that can help you to remember “PEMDAS,” which is the correct order in which to solve a problem.

Let’s practice with a sample problem and solve it step by step:

*12 + (2 x 5) x 3⁴ ÷ 9*

**Parentheses:**12 + (10) x 3⁴ ÷ 9**Exponents:**12 + (10) x 81 ÷ 9**Multiplication/Division:**12 + 810 ÷ 9

= 12 + 90**Addition/Subtraction:**102

Remember to solve problems left to right, just like you read from left to right. Notice that in the multiplication/division step, we multiplied (left) before we divided (right).

It’s very important to practice using the order of operations–this concept will come in handy for lots of problems on the Elementary Algebra test!

**Roots and Exponents**

You’re already somewhat familiar with exponents from our practice. A root can be thought of as the opposite of an exponent. For example:

5² = 5 and √25= 5

The sign for a square root is √. This sign tells you to find what number multiplied by itself is equivalent to a number under the sign.

3³ = 27 and ∛27= 3

The sign for a cube root is ∛. This sign tells you to find what number multiplied by itself three times is equivalent to the number under the sign.

**Working with Exponents**

Let’s take a look at some general rules you will need to know to work with exponents.

**The product rule:**If two like bases are multiplied, add the exponents. Example: 4³ · 4² = 4⁵ = 4 · 4 · 4 · 4 · 4 = 1,024**The quotient rule:**If a base is divided by a like base, subtract the exponents. Example: 3⁵ ÷ 3³ = 3² = 9**Negative exponents:**Any nonzero number raised to a negative power is equal to the reciprocal raised to the opposite power. Example: 6⁻² = 1/36.**The zero exponent rule:**Any nonzero number raised to the zero power is 1. Example: 9⁰ = 1.

Let’s solve the problem below:

x⁵(x + y²) =

x⁶ + x⁵y²

**Monomials**

A monomial is a number, a variable, or the product of a number and a variable that has only whole exponents. Remember that a variable, such as *x, *stands in place of a value.

Here are some examples of monomials:

9x, 14*⁴, *14 / 2*,* x²

**Adding and Subtracting Monomials**

Now we will practice adding and subtracting some monomials.

- 10y – 7y = 3y
- 17x + 8x -3x – (-3x) = 22x – (-3x) = 25x
- 11c² + 12c² + 4c² = 27c²

**Multiplying and Dividing Monomials**

Next, let’s practice multiplying and dividing monomials.

**Polynomials**

A polynomial is the sum of two or more monomials. Here is an example:

14x² – 12x + 10

**Adding and Subtracting Polynomials**

Let’s practice some addition and subtraction with polynomials.

2x² + 6x + 5 + 3x² – 2x – 1

First, place the like terms together.

2x² + 3x² + 6x − 2x + 5 − 1

Next, add (or subtract) the like terms:

5x² + – 4x + 4 or 5x² – 4x + 4

**Multiplying and Dividing Polynomials**

(x + 2y)(3x − 4y + 5)

First, multiply each term in the parentheses to the right by x. Then, multiply each term in the parentheses to the right by 2y.

3x² − 4xy + 5x + 6xy − 8y² + 10y

Once again, you need to add like terms. In this case 6xy and -4xy are added:

3x² + 2xy + 5x − 8y² + 10y

**Equations and Inequalities**

On the Elementary Algebra test, you will also need to use your math skills to work with equations and inequalities. Let’s look at some concepts that are likely to appear on the test.

**Linear Equations and Inequalities**

A linear equation has an equal sign and linear expressions. We will review graphing linear equations later on. For now, let’s solve the problem below:

*6x + 15 = 2x + 25*

The equals sign tells you that the values on each side are equivalent. If we are going to solve for x, we need to combine like terms. Let’s move 15 to the other side of the equation.

When we move 15 to the other side, we need to use its inverse (-15).

*6x + 15 = 2x + 25*

-15 -15

__________

6x = 2x + 10

Next, we need to get all of our x variables on one side of the equation. We need to subtract *2x* from the left side of the equation:

*6x = 2x + 10*

*-2x -2x*

_________

*4x = 10*

* x = *10/4 *or *5/2

A linear inequality is solved the same way. However, instead of an equal sign (=), linear inequalities have a greater than sign (﹥), a greater than or equal to sign (≥), a less than sign (﹤), or a less than or equal to sign (≤).

**Quadratic Equations**

A quadratic equation describes a parabola, or curve, when it is graphed. Quadratic equations contain squares and their standard form is expressed as

*ax*² *+ bx + c = 0*.

Sometimes, quadratic equations are not in standard form. Let’s look at an example and simplify it so that it is in standard form:

½ *[(184t**2** – 166t**2**) + (-18t x 2)] + 2 = -14t +7*

Let’s complete the simple arithmetic in the brackets:

½ *(18t**2** – 36t) + 2 = -14t +7*

Now, let’s multiply the variables in parentheses by ½:

*9t**2** – 18t + 2 = -14t + 7*

We need to combine like terms next. Move -14t to the other side of the equation via addition:

*2t**2** – 4t + 2 = 7*

Next, we need to subtract 7 from each side in order for our equation to equal 0:

*2t**2** -4t – 5 = 0*

*ax**2** + bx + c = 0*

So, *a = 2, b = -4, *and* c = −5*. You’re all done!

And that’s some basic info about the Elementary Algebra test.

## Mathematics Test: College-Level Math

The College-Level Math test has 20 multiple-choice questions. You will have an unlimited amount of time to complete this test.

It will be very important to be familiar with the following concepts as you take the test:

- Rational Expressions
- Systems of Equations
- Coordinate Geometry
- Sequences
- Logarithms and Exponential Equations
- Permutations
- Trigonometric Functions

So, let’s talk about them.

**Rational Expressions**

Rational expressions are fractions with a polynomial in the numerator, denominator, or both. Rational expressions can be simplified just like numeric fractions can be simplified.

**Simplifying a Rational Expression**

**Systems of Equations**

On the College-Level Math CAT, you are likely to encounter problems that ask you to work with systems of equations. Systems of equations are two or more equations that have variables in common. They can also be described as two or more linear equations that work together.

Here is an example of two linear equations that form a system of equations:

x + y = 6

−3x + y = 2

First, subtract the second equation from the first equation:

x + y − (−3x + y) = 6 − 2

x + y + 3x − y = 6 − 2

4x = 4

Now, simplify to find the x-value:

x = 1

Next, find the value of *y* by plugging the* x* value into one of the equations:

x + y = 6

1 + y = 6

y = 5

So, x = 1 and y = 5.

**Coordinate Geometry**

The test will probably ask you questions related to coordinate geometry, which refers to geometric calculations that can be described on a coordinate plane. An example of a coordinate plane is shown below:

Points on a graph are expressed as (x, y). The *x* value describes where the point is on the horizontal x-axis and the *y* value describes where the point is on the vertical y-axis.

**Lines and Slope**

The formula y = mx + b describes a line on a coordinate plane. The *x *and *y* values of any point on the line may be substituted in the equation. The variable m is the slope and b is the y-intercept. The y-intercept is where the line crosses the y-axis.

Let’s look at an example:

At what point does a line passing through point K (5, –2) and point L (3, 4) intersect the y-axis of a coordinate plane?

This question is asking you to find the y-intercept (*b*) of the line. You already know the values of two points, so you just need to find the slope (m).

Now you have the slope. Go back to the formula y = mx + b. You can use either point to find the answer. Let’s plug in the values from point L:

y = mx + b

4 = (-3)3 + b

4 = -9 + b

13 = b

So, you can determine that this line crosses the y-axis at (0,13).

**Graphing Functions**

In order to score well on the test, you will more than likely need to have an understanding of how functions are graphed. When graphing functions, you can think of *f(x)* as *y*.

Here is an example function: f(x) = 1 / x

You can form a table by plugging in some x values and solving for f(x):

Your graph will look like this:

**Sequences**

Sometimes, you will be given a sequence of numbers and asked to find the next number in the sequence, or the nth term (another number in the sequence). Other times, you may be asked to find the common difference between the numbers in a sequence.

Take a look at the sequence below:

1, 3, 9, 27, 81

If you were asked to find the next number in the sequence, 243 would be the correct answer. Each number is 3 times the value of the number before it.

Here’s another sequence:

6, 16, 11, 21, 16, 26, 21

The next term in this sequence would be 31. This is because the sequence is add 10, subtract 5, add 10, subtract 5, over and over again.

**Logarithms and Exponential Equations**

A logarithm is the power to which a number must be raised in order to get another number.

**Logs**

Consider the following expression:

log₆ 36 =

This means the same thing as “How many 6s must be multiplied to get 36?”

log₆ 36 = 2

You know that 6 x 6 = 36. The number 6 appears twice.

Likewise, log₅ 125 = 3. This is because 5 x 5 x 5 = 125. The number 5 appears three times.

There are a few other examples below.

log₃ 81 = 4 because 3⁴ = 81

log₂ 32 = 5 because 2⁵ = 32

log₈ 4,096 = 4 because 8⁴ = 4,096

**Solving Logarithmic Equations**

Let’s look at an equation in which the bases of the logarithms on either side are the same:

logᵣx = logᵣy

In the given equation, you can determine that *x = y* since the logs are the same. Notice that you are just dropping the “logᵣ.” Here is an example using real numbers:

log₉ (√x)² = log₉(6x – 1)

Therefore, (√x)² = (6x – 1)

x = 6x -1

+1 +1

________

x + 1 = 6x

x + 1 = 6x

-1x -1x

________

1 = 5x

1 = 5x

÷5 ÷5

______

⅕= x

In the given equation, x = ⅕. You can check the answer by substituting ⅕ for x in the original equation.

log₉ (⅕)² = log₉(6⅕- 1) or (⅕)² = (6⅕- 1) = 1/25

**Solving Exponential Equations**

Sometimes, logarithms come in handy when you are presented with an exponential equation. Look at the following example:

5 x 2ˣ = 240

When you divide both sides by 5, you get:

2ˣ = 48

Now, let’s convert the equation to a logarithmic form:

2ˣ = 48 is the same as log₂48 = x

In this equation, x = log₂.

Let’s try solving another exponential equation using logarithms.

6 x 10²ˣ = 48

Divide by 6 to get 10²ˣ = 8.

Convert the equation to logarithmic form: log₁₀8 = 2x

Divide by 2 to get:

**Permutations**

Permutations describe how many unique lists can be created. In a permutation, the order of the lists matter and the same item may not appear twice.

For example, if there is a bagel (B), a bowl of soup (S), and a piece of fruit (F) on the table and Serena eats one item, and then a second item, there is a one in ___ chance that she eats the bagel, then the soup.

First, let’s find all of the possible combinations:

BS, SB, BF, FB, SF, FS

There are six possible combinations. The chance that she will eat the bagel, then the soup (the first combination above) is one in six.

The permutation notation for this problem would be 3P2.

In this formula, k = known elements and n = the subset being described.

Try putting the following problem in permutation notation:

How many three-letter combinations are possible using the 26 letters of the English alphabet?

The permutation notation for this problem is 26P3.

The n! is called the “n factorial” and it is calculated by multiplying an integer, n, with each integer that precedes it, all the way down to 1. (For example, 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.)

Notice that you do not need to describe anything beyond 23! in the dividend (the upper half of the equation). This is because the divisor is 23! and 23! will cancel 23! out.

**Trigonometric Functions**

On the test, you are likely to need to perform some basic trigonometry. Let’s review some concepts.

**Evaluating Trigonometric Functions**

Some of the most commonly used trigonometric functions are sine, cosine, and tangent. Take a look at the figure below and the corresponding table that describes each of these functions.

The hypotenuse is the side opposite the right angle, the “opposite” side is opposite from the angle of interest (angle q in the example), and the adjacent side is the side which connects the angle of interest to the right angle.

So, how might this concept appear on the CAT? Take a look at the sample question below:

In this triangle, what equation describes the sine 60°?

All you need to do is to plug the values into the formula for sine. We know the hypotenuse and the opposite side, just not the adjacent side.

Sine(angle) = opposite ÷ hypotenuse

Sine(60) = 3 ÷ 5

Sine(60) = .6