# 5.E: Review Exercises and Sample Exam - Mathematics

## Review Exercises

Exercise (PageIndex{1}) Rules of Exponents

Simplify.

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

Evaluate.

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

Multiply.

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

Divide.

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

Simplify.

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) Answer 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}) Answer 2. (5.796×10^{19}) 4. (2.25×10^{−20}) 6. (2.5×10^{−18}) ## Simple Exam Exercise (PageIndex{12}) Simplify. 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}) Answer 1. (−10x^{5}y) 3. (4x^{2}y^{5}) Exercise (PageIndex{13}) Evaluate. 1. (2x^{2}−x+5), where (x=−5) 2. (a^{2}−b^{2}), where (a=4) and (b=−3) Answer 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}) Answer 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}) Simplify. 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.

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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.