5.E: Review Exercises and Sample Exam - Mathematics

Review Exercises

Exercise (PageIndex{1}) Rules of Exponents


  1. (7^{3}⋅7^{6})
  2. (5^{9}5^{6})
  3. (y^{5}⋅y^{2}⋅y^{3})
  4. (x^{3}y^{2}⋅xy^{3})
  5. (−5a^{3}b^{2}c⋅6a^{2}bc^{2})
  6. (frac{55x^{2}yz}{55xyz^{2}})
  7. ((frac{−3a^{2}b^{4}}{2c^{3}})^{2})
  8. ((−2a^{3}b^{4}c^{4})^{3})
  9. (−5x^{3}y^{0}(z^{2})^{3}⋅2x^{4}(y^{3})^{2}z)
  10. ((−25x^{6}y^{5}z)^{0})
  11. Each side of a square measures (5x^{2}) units. Find the area of the square in terms of (x).
  12. Each side of a cube measures (2x^{3}) units. Find the volume of the cube in terms of (x).

1. (7^{9})

3. (y^{10})

5. (−30a^{5}b^{3}c^{3})

7. (frac{9a^{4}b^{8}}{4c^{6}})

9. (−10x^{7}y^{6}z^{7})

11. (A=25x^{4})

Exercise (PageIndex{2}) Introduction to Polynomials

Classify the given polynomial as a monomial, binomial, or trinomial and state the degree.

  1. (8a^{3}−1)
  2. (5y^{2}−y+1)
  3. (−12ab^{2})
  4. (10)

1. Binomial; degree (3)

3. Monomial; degree (3)

Exercise (PageIndex{3}) Introduction to Polynomials

Write the following polynomials in standard form.

  1. (7−x^{2}−5x)
  2. (5x^{2}−1−3x+2x^{3})

1. (-x^{2}-5x+7)

Exercise (PageIndex{4}) Introduction to Polynomials


  1. (2x^{2}−x+1), where (x=−3)
  2. (frac{1}{2}x−frac{3}{4}), where (x=frac{1}{3})
  3. (b^{2}−4ac), where (a=−frac{1}{2}, b=−3), and (c=−frac{3}{2})
  4. (a^{2}−b^{2}), where (a=−frac{1}{2}) and (b=−frac{1}{3})
  5. (a^{3}−b^{3}), where (a=−2) and (b=−1)
  6. (xy^{2}−2x^{2}y), where (x=−3) and (y=−1)
  7. Given (f(x)=3x^{2}−5x+2), find (f(−2)).
  8. Given (g(x)=x^{3}−x^{2}+x−1), find (g(−1)).
  9. The surface area of a rectangular solid is given by the formula (SA=2lw+2wh+2lh), where (l, w), and (h) represent the length, width, and height, respectively. If the length of a rectangular solid measures (2) units, the width measures (3) units, and the height measures (5) units, then calculate the surface area.
  10. The surface area of a sphere is given by the formula (SA=4πr^{2}), where (r) represents the radius of the sphere. If a sphere has a radius of (5) units, then calculate the surface area.

1. (22)

3. (6)

5. (−7)

7. (f(−2)=24)

9. (62) square units

Exercise (PageIndex{5}) Adding and Subtracting Polynomials

Perform the operations.

  1. ((3x−4)+(9x−1))
  2. ((13x−19)+(16x+12))
  3. ((7x^{2}−x+9)+(x^{2}−5x+6))
  4. ((6x^{2}y−5xy^{2}−3)+(−2x^{2}y+3xy^{2}+1))
  5. ((4y+7)−(6y−2)+(10y−1))
  6. ((5y^{2}−3y+1)−(8y^{2}+6y−11))
  7. ((7x^{2}y^{2}−3xy+6)−(6x^{2}y^{2}+2xy−1))
  8. ((a^{3}−b^{3})−(a^{3}+1)−(b^{3}−1))
  9. ((x^{5}−x^{3}+x−1)−(x^{4}−x^{2}+5))
  10. ((5x^{3}−4x^{2}+x−3)−(5x^{3}−3)+(4x^{2}−x))
  11. Subtract (2x−1) from (9x+8).
  12. Subtract (3x^{2}−10x−2) from (5x^{2}+x−5).
  13. Given (f(x)=3x^{2}−x+5) and (g(x)=x^{2}−9), find ((f+g)(x)).
  14. Given (f(x)=3x^{2}−x+5) and (g(x)=x^{2}−9), find ((f−g)(x)).
  15. Given (f(x)=3x^{2}−x+5) and (g(x)=x^{2}−9), find ((f+g)(−2)).
  16. Given (f(x)=3x^{2}−x+5) and (g(x)=x^{2}−9), find ((f−g)(−2)).

1. (12x−5)

3. (8x^{2}−6x+15)

5. (8y+8)

7. (x^{2}y^{2}−5xy+7)

9. (x^{5}−x^{4}−x^{3}+x^{2}+x−6)

11. (7x+9)

13. ((f+g)(x)=4x^{2}−x−4)

15. ((f+g)(−2)=14)

Exercise (PageIndex{6}) Multiplying Polynomials


  1. (6x^{2}(−5x^{4}))
  2. (3ab^{2}(7a^{2}b))
  3. (2y(5y−12))
  4. (−3x(3x^{2}−x+2))
  5. (x^{2}y(2x^{2}y−5xy^{2}+2))
  6. (−4ab(a^{2}−8ab+b^{2}))
  7. ((x−8)(x+5))
  8. ((2y−5)(2y+5))
  9. ((3x−1)^{2})
  10. ((3x−1)^{3})
  11. ((2x−1)(5x^{2}−3x+1))
  12. ((x^{2}+3)(x^{3}−2x−1))
  13. ((5y+7)^{2})
  14. ((y^{2}−1)^{2})
  15. Find the product of (x^{2}−1) and (x^{2}+1).
  16. Find the product of (32x^{2}y) and (10x−30y+2).
  17. Given (f(x)=7x−2) and (g(x)=x^{2}−3x+1), find ((f⋅g)(x)).
  18. Given (f(x)=x−5) and (g(x)=x^{2}−9), find ((f⋅g)(x)).
  19. Given (f(x)=7x−2) and (g(x)=x^{2}−3x+1), find ((f⋅g)(−1)).
  20. Given (f(x)=x−5) and (g(x)=x^{2}−9), find ((f⋅g)(−1)).

1. (−30x^{6})

3. (10y^{2}−24y)

5. (2x^{4}y^{2}−5x^{3}y^{3}+2x^{2}y)

7. (x^{2}−3x−40)

9. (9x^{2}−6x+1)

11. (10x^{3}−11x^{2}+5x−1)

13. (25y^{2}+70y+49)

15. (x^{4}−1)

17. ((f⋅g)(x)=7x^{3}−23x^{2}+13x−2)

19. ((f⋅g)(−1)=−45)

Exercise (PageIndex{7}) Dividing Polynomials


  1. (frac{7y^{2}−14y+28}{7})
  2. (frac{12x^{5}−30x^{3}+6x}{6x})
  3. (frac{4a^{2}b−16ab^{2}−4ab}{−4ab})
  4. (frac{6a^{6}−24a^{4}+5a^{2}}{3a^{2}})
  5. ((10x^{2}−19x+6)÷(2x−3))
  6. ((2x^{3}−5x^{2}+5x−6)÷(x−2) )
  7. (frac{10x^{4}−21x^{3}−16x^{2}+23x−20}{2x−5})
  8. (frac{x^{5}−3x^{4}−28x^{3}+61x^{2}−12x+36}{x−6})
  9. (frac{10x^{3}−55x^{2}+72x−4}{2x−7})
  10. (frac{3x^{4}+19x^{3}+3x^{2}−16x−11}{3x+1})
  11. (frac{5x^{4}+4x^{3}−5x^{2}+21x+21}{5x+4})
  12. (frac{x^{4}−4}{x−4})
  13. (frac{2x^{4}+10x^{3}−23x^{2}−15x+30}{2x^{2}−3})
  14. (frac{7x^{4}−17x^{3}+17x^{2}−11x+2}{x^{2}−2x+1})
  15. Given (f(x)=x^{3}−4x+1) and (g(x)=x−1), find ((f/g)(x)).
  16. Given (f(x)=x^{5}−32) and (g(x)=x−2), find ((f/g)(x)).
  17. Given (f(x)=x^{3}−4x+1) and (g(x)=x−1), find ((f/g)(2)).
  18. Given (f(x)=x^{5}−32) and (g(x)=x−2), find ((f/g)(0)).

1. (y^{2}−2y+4)

3. (−a+4b+1)

5. (5x−2)

7. (5x^{3}+2x^{2}−3x+4)

9. (5x^{2}−10x+1+frac{3}{2x−7})

11. (x^{3}−x+5+frac{1}{5x+4})

13. (x^{2}+5x−10)

15. ((f/g)(x)=x^{2}+x−3−frac{2}{x−1})

17. ((f/g)(2)=1)

Exercise (PageIndex{8}) Negative Exponents


  1. ((−10)^{−2})
  2. (−10^{−2})
  3. (5x^{−3})
  4. ((5x)^{−3})
  5. (frac{1}{7y^{-3}})
  6. (3x^{−4}y^{−2})
  7. (frac{−2a^{2}b^{−5}}{c^{−8}})
  8. ((−5x^{2}yz^{−1})^{−2})
  9. ((−2x^{−3}y^{0}z^{2})^{−3})
  10. ((frac{−10a^{5}b^{3}c^{2}}{5ab^{2}c^{2}})^{−1})
  11. ((frac{a^{2}b^{−4}c^{0}}{2a^{4}b^{−3}c})^{−3})

1. (frac{1}{100})

3. (frac{5}{x^{3}})

5. (frac{y^{3}}{7})

7. (frac{−2a^{2}c^{8}}{b^{5}})

9. (frac{−x^{9}}{8z^{6}})

11. (8a^{6}b^{3}c^{3})

Exercise (PageIndex{9}) Negative Exponents

The value in dollars of a new laptop computer can be estimated by using the formula (V=1200(t+1)^{−1}), where (t) represents the number of years after the purchase.

  1. Estimate the value of the laptop when it is (1frac{1}{2}) years old.
  2. What was the laptop worth new?

2. $(1,200)

Exercise (PageIndex{10}) Negative Exponents

Rewrite using scientific notation.

  1. (2,030,000,000)
  2. (0.00000004011)

2. (5.796×10^{19})

Exercise (PageIndex{11}) Negative Exponents

Perform the indicated operations.

  1. ((5.2×10^{12})(1.8×10^{−3}))
  2. ((9.2×10^{−4})(6.3×10^{22}))
  3. (frac{4×10^{16}}{8×10^{−7}})
  4. (frac{9×10^{−30}}{4×10^{−10}})
  5. (5,000,000,000,000 × 0.0000023)
  6. (frac{0.0003}{120,000,000,000,000})

2. (5.796×10^{19})

4. (2.25×10^{−20})

6. (2.5×10^{−18})

Simple Exam

Exercise (PageIndex{12})


  1. (−5x^{3}(2x^{2}y))
  2. ((x^{2})^{4}⋅x^{3}⋅x)
  3. (frac{(−2x^{2}y^{3})^{2}}{x^{2}y})
    1. ((−5)^{0})
    2. (−5^{0})

1. (−10x^{5}y)

3. (4x^{2}y^{5})

Exercise (PageIndex{13})


  1. (2x^{2}−x+5), where (x=−5)
  2. (a^{2}−b^{2}), where (a=4) and (b=−3)

1. (60)

Exercise (PageIndex{14})

Perform the operations.

  1. ((3x^{2}−4x+5)+(−7x^{2}+9x−2) )
  2. ((8x^{2}−5x+1)−(10x^{2}+2x−1) )
  3. ((frac{3}{5}a−frac{1}{2})−(frac{2}{3}a^{2}+frac{2}{3}a−frac{2}{9})+(frac{1}{15}a−frac{5}{18}))
  4. (2x^{2}(2x^{3}−3x^{2}−4x+5))
  5. ((2x−3)(x+5))
  6. ((x−1)^{3})
  7. (frac{81x^{5}y^{2}z}{-3x^{3}yz})
  8. (frac{10x^{9}−15x^{5}+5x^{2}}{−5x^{2}})
  9. (frac{x^{3}−5x^{2}+7x−2}{x−2})
  10. (frac{6x^{4}−x^{3}−13x^{2}−2x−1}{2x−1})

1. (−4x^{2}+5x+3 )

3. (−frac{2}{3}a^{2}−frac{5}{9})

5. (2x^{2}+7x−15 )

7. (−27x^{2}y)

9. (x^{2}−3x+1)

Exercise (PageIndex{15})


  1. (2^{−3})
  2. (−5x^{−2})
  3. ((2x^{4}y^{−3}z)^{−2})
  4. ((frac{−2a^{3}b^{−5}c^{−2}}{ab^{−3}c^{2}})^{−3})
  5. Subtract (5x^{2}y−4xy^{2}+1) from (10x^{2}y−6xy^{2}+2).
  6. If each side of a cube measures (4x4) units, calculate the volume in terms of (x).
  7. The height of a projectile in feet is given by the formula (h=−16t^{2}+96t+10), where (t) represents time in seconds. Calculate the height of the projectile at (1frac{1}{2}) seconds.
  8. The cost in dollars of producing custom t-shirts is given by the formula (C=120+3.50x), where (x) represents the number of t-shirts produced. The revenue generated by selling the t-shirts for $(6.50) each is given by the formula (R=6.50x), where (x) represents the number of t-shirts sold.
    1. Find a formula for the profit. (profit = revenue − cost)
    2. Use the formula to calculate the profit from producing and selling (150) t-shirts.
  9. The total volume of water in earth’s oceans, seas, and bays is estimated to be (4.73×10^{19}) cubic feet. By what factor is the volume of the moon, (7.76×10^{20}) cubic feet, larger than the volume of earth’s oceans? Round to the nearest tenth.

1. (frac{1}{8})

3. (frac{y^{6}}{4x^{8}z^{2}})

5. (5x^{2}y−2xy^{2}+1)

7. (118) feet

9. (16.4)

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What Is the PERT Exam?

The PERT test is also known by its full name: the Postsecondary Education Readiness Test. It is meant to gauge your preparedness for the college environment based on your capabilities with three core subjects: Writing, Reading, and Mathematics. You have likely gained ample experience with all three of these subjects throughout your high school career, both within your classes and on other standardized tests you have taken previously.

The PERT exam will be similar to past standardized tests you’ve taken, save for one detail: there is no real scoring system for this exam. More specifically, there is no such thing as failing. Instead, you should think of PERT passing scores as a rubric of sorts. It is meant to let you know where your weaknesses and strengths lie so you can organize your college classes accordingly.

You do not have to worry about time limits as you take the exam. Take all the time you need to complete the test as thoroughly and accurately as you can. The test will be presented entirely in multiple choice format, and is administered digitally.

As stated above, the PERT exam splits off into three core subjects. The remainder of this page will cater to information pertaining to the PERT Math test.

What is the TSIA2?

The Texas Success Initiative Assessment 2.0 (TSIA2) is the revised version of the TSIA1, updated and improved to support student success through effective testing and guidance. The Texas Success Initiative requires all Texas public institutions of higher education to determine their students’ readiness for success in freshman-level academic courses. All students entering Texas public colleges, technical schools, and universities undergo assessment using the TSIA2, unless exempt.

The TSIA2 consists of three examinations – Mathematics, English Language Arts and Reading (ELAR), and Essay. Students are placed into appropriate coursework based on their results from the TSIA2. This article discusses the Mathematics suite of the TSIA2.

You can’t fail the PERT, but there are serious repercussions to doing poorly on the test. A low score will put you into remedial courses, which is not a place you want to be. These courses don’t even give you credit toward your degree, so you can end up wasting lots of time and money at the very start of your college career. Therefore, while you can’t “fail” the PERT, you should be very concerned about what happens if you get a low score.

The PERT math test is hard simply because many students struggle with math. While the math on the PERT is not as hard as it is on the ACT or SAT, for example, it’s challenging enough to send most students down to remedial classes unless they take the time to prepare. Scores range from 50 to 150, and a 114 or more is required to place out of remedial math.

TEAS Study Guide

It’s important that you brush up on your basic (9-12 grade) reading, math, science, English and language skills and take a ATI TEAS Practice Test or two (or 10). It’s also a good idea to time yourself since the actual exam is timed.

ATI also offers products to help you study. They include the ATI TEAS Study Package, ATI TEAS SmartPrep Study Package, ATI TEAS Study Manual – Sixth Edition, and the ATI TEAS Practice Test Assessment Packages. Package Includes:

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  • ATI TEAS Study Manual
  • ATI TEAS Online Practice Form A
  • ATI TEAS Online Practice Form B

Free Online Study Guide:

Can I use a calculator on the ATI TEAS?

  • Students are not allowed to bring a calculator. The ATI TEAS will have an embedded calculator within the online test that students can use.

How long will it take for me to get my TEAS scores?

How do I know if I passed the ATI TEAS?

It depends on the College/Institution, in general, the minimum ATI TEAS score varies from (55% to 75%). You must achieve this score in your first two attempts at the ATI TEAS. If you cannot achieve a score of the minimum score in your first two attempts, then you may not be considered for the submit an application to the respective colleges. Admission to the Nursing Program is on a competitive basis determined by TEAS exam scores.

How do I get my official TEAS transcript sent to another college?

For a fee, ATI will electronically transmit your results to the school of your choice. After you have your results, log onto the ATI website and request a TEAS transcript.

Free Calculus 1 Diagnostic Tests

Calculus I courses provide students with an in-depth introduction to the core concepts of limits, derivatives, and integrals, building on the preliminary understanding of these concepts that students gained in Pre-Calculus courses while preparing them for the more advanced material of Calculus II, Calculus II, and Differential Equations. Calculus courses are often necessary for students to be able to tackle not only these higher-level mathematics courses, but advanced material in the sciences. If you are considering majoring in math, science, or any other quantitative field, taking Calculus before reaching college can be a real boon, as high school Calculus courses often take the material at a somewhat slower pace than collegiate courses, making sure that students fully understand each concept before moving on. Whether you need top Calculus tutors in New York, Calculus tutors in Chicago, or top Calculus tutors in Los Angeles, working with a pro may take your studies to the next level.

The material taught in Calculus I courses can be broken down into three main categories: limits, derivatives, and integrals however, most Calculus I courses begin with a review of the basic features of functions graphed on the coordinate plane, including continuity, asymptotes, and absolute and local extrema. Students may be asked to find the slope of a line or slope at a point when reviewing these concepts.

When delving into the concept of function limits, Calculus courses typically begin with the process of calculating and estimating simple limits and proceed to introduce concepts of asymptotes and continuity, calculating limits to infinity, and other complexities.

Discussing derivatives is a major part of every Calculus I course. Students are introduced to derivatives through discussions of the definition of a derivative, the limit definition of a derivative, and differential equations in order to bolster students&rsquo conceptual understandings of derivatives. It is crucial that students fully understand what derivatives represent as they progress in Calculus I, as they are soon asked to apply this knowledge by calculating derivatives at a point and of a function, as well as second derivatives. They are also taught the Chain Rule. Students are also asked to graph derivatives and second derivatives, along with linear approximations of derivatives. As student knowledge of derivatives increases, Calculus I introduces the concepts of increasing and decreasing intervals, concavity and convexity, points of inflection, and slope fields. Students are also asked to make use of the Mean Value Theorem. Specific derivatives, such as the derivatives of logarithms, exponents, sums, quotients, products, and trigonometric functions, are taught, as well as implicit differentiation.

The last major topic of every Calculus I course is integrals. Integrals are introduced by talking about the definition of an integral, integral notation, definite integrals, and Riemann sums. After the concept of an integral is introduced in detail, students are taught the Fundamental Theorem of Calculus, how to take the integral of a function, and how to graph integrals. Increasingly difficult problems are likely to appear, as students are asked to take the integral of more complex functions such as sums, quotients, and products, logarithms, exponents, and trigonometric functions. Varsity Tutors offers resources like free Calculus 1 Diagnostic Tests to help with your self-paced study, or you may want to consider an Calculus 1 tutor.

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College Mathematics

The following College Mathematics sample questions don't appear on an actual CLEP examination. They are intended to give potential test takers an indication of the format and difficulty level of the examination and to provide content for practice and review. Knowing the correct answers to all of the sample questions isn't a guarantee of satisfactory performance on the exam.

Directions: An online scientific calculator will be available for the questions on this test. For each of the questions below, select the BEST of the choices given.

  1. Unless otherwise specified, the domain of any function f is assumed to be the set of all real numbers x for whichf (x)is a real number.
  2. Figures that accompany questions are intended to provide information useful in answering the questions. The figures are drawn as accurately as possible EXCEPT when it is stated in a specific question that the figure is not drawn to scale.
  3. If a principal of P dollars is invested at an annual interest rate r, compounded n times per year, and no further withdrawals or deposits are made to the account, then the future value A, the account balance after t years, is given by the formula
  4. If a principal of P dollars is invested at an annual interest rate r, compounded continuously, and no further withdrawals or deposits are made to the account, then the future value A, the account balance after t years, is given by the formula A=Pe rt .
  5. At an interest rate r, compounded n times per year, the effective annual yield or annual percentage rate (APR), is given by the formula .

To learn about the exam directions and format, go to the College Mathematics exam page. For more sample questions and info about the test, see the CLEP Official Study Guide.

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  1. Use Diagrams / Models
  2. Act it Out
  3. Use Before & After
  4. Use Systematic Listing
  5. Look for Patterns
  6. Work Backwards
  7. Use Guess & Check
  8. Simplify the Problem
  9. Make Supposition
  10. Solve Part of the Problem
  11. Paraphrase the Problem

To help the students preparing for Primary 1 to Primary 6 exams including Primary School Leaving Exam (PSLE), we provide interactive online practice tests and excllent free mathematics worksheets /test papers in PDF for download here.