Operations With Real Numbers and Expressions
Name_______________________
1. Select the correct symbol.
?
25
/3
A. >
B. =
C. <
2. Evaluate the following expression when n = 6.
4|2 - 5n| + |4|
A. -116
B. 116
C. 108
D. -108
3. Jessica is taking a road trip from California to Washington. Every 4 hours, she travels between 208 and 248
miles. Twelve hours ago, Jessica had traveled 96 miles from her starting point in California. Which is a
reasonable measure of Jessica's distance from her starting point in California now?
A. 228 miles
B. 780 miles
C. 588 miles
D. 324 miles
4. Simplify the following expression.
A.
B.
C.
D.
5. Simplify.
A.
B.
C.
D.
6. Simplify: (6x4 + 2x3 + 3x + 7) + (3x4 - 9x3 - 2x - 15)
A. 9x4 - 7x3 - 8
B. 9x4 - 7x3 + x - 8
C. 9x4 - 7x3 + x + 8
D. 3x4 - 7x3 + x - 8
7. Simplify.
A.
B.
C.
D.
8. The Rose Theater seats 716 people, and the theater is hosting a play for 4 nights. If the theater is at about
51% capacity for each night of the play, approximately how many people will attend the play?
A.
B.
C.
D.
9. Factor the following expression completely.
4x2 - 100
A. (4x + 5)(x - 5)
B. 4(x + 5)(x - 5)
C. 4(x - 5)2
D. 4(x2 - 25)
10. Simplify the following expression.
A.
B.
C.
D.
11. Simplify: 5(12 - |-10 + 5|) - |12 - 2|2
A. -115
B. -45
C. -465
D. -65
12. Order the following from least to greatest.
A.
B.
C.
D.
13. Factor the following expression completely.
x4 - 1
A. (x - 1)(x + 1)(x - 1)(x + 1)
B. (x - 1)(x3 + 1)
C. (x - 1)(x + 1)(x2 + 1)
D. (x2 - 1)(x2 + 1)
14.
14x3y2z3
10x3y4
What is the greatest common factor (GCF) of the monomials shown above?
A. 2x3y4z3
B. 70x6y6z3
C. 2x3y2
D. 70x3y4z3
15. Simplify: (6x2 + 3x + 8) + (4x2 - 8x - 3)
A. 2x2 - 5x + 11
B. 10x2 - 11x - 5
C. 10x2 - 5x + 5
D. 10x2 - 11x + 5
16. Look at the two monomials below.
9u3v4w3
5uv4w2
What is the least common multiple (LCM) of the monomials shown above?
A. u4v8w5
B. 45u3v4w3
C. 45u4v8w5
D. uv4w2
17. Order the following numbers from greatest to least.
A.
B.
C.
D.
18.
9xy2
33xy
What is the greatest common factor (GCF) of the monomials shown above?
A. 3xy
B. 99xy2
C. 99x2y3
D. 3xy2
19. Simplify the following expression.
A.
B.
C.
D.
20. Simplify.
A.
B.
C.
D.
Answers
1. A
2. B
3. B
4. C
5. A
6. B
7. C
8. C
9. B
10. B
11. D
12. C
13. C
14. C
15. C
16. B
17. D
18. A
19. C
20. D
Explanations
1. In order to compare the two numbers, write both of them in decimal form.
8.88...
25
/3
So,
> 25/3.
2. Absolute value is the magnitude of a number irrespective of its sign.
Evaluate the given expression piece-by-piece, using order of operations:
|2 - 5n| = |2 - 5 × 6| = |2 - 30| = |-28| = 28
4 × 28 = 112
|4| = 4
112 + 4 = 116
3. Every 4 hours, Jessica's distance from her starting point in California increases between 208 and 248 miles.
Find how many 4 hour intervals Jessica has traveled in 12 hours by dividing.
Find the range of increase in distance.
208 miles × 3 intervals = 624 miles
248 miles × 3 intervals = 744 miles
Jessica was already 96 miles from her staring point in California; therefore, add 96 to the range values.
624 miles + 96 miles = 720 miles
744 miles + 96 miles = 840 miles
Jessica has traveled between 720 miles and 840 miles from her starting point in California.
The only answer choice in this range is 780 miles.
4. When dividing a polynomial by a monomial, factor to find common multiples and simplify.
5. Even though 48 is not a perfect square,
another number.
can be simplified by factoring 48 into a perfect square and
6. Since it is addition of two polynomials, drop the parenthesis and collect like terms.
(6x4 + 2x3 + 3x + 7) + (3x4 - 9x3 - 2x - 15) = 6x4 + 2x3 + 3x + 7 + 3x4 - 9x3 - 2x - 15
= (6x4 + 3x4) + (2x3 - 9x3) + (3x - 2x) + (7 - 15)
= 9x4 - 7x3 + x - 8
7. Even though 80 is not a perfect square,
another number.
can be simplified by factoring 80 into a perfect square and
8. Calculate the approximate number of seats that will be occupied per night.
Calculate the approximate number of people who will attend the play.
Approximately 1,460 people will attend the play.
9. First, factor out 4, then factor using the difference of two squares.
4x2 - 100 = 4(x2 - 25)
= 4(x + 5)(x - 5)
10. First, factor the numerator.
Now, replace the numerator with the factored form and cancel out the common factors.
11. Use the order of operations to simplify the expression.
5(12 - |-10 + 5|) - |12 - 2|2 = 5(12 - |-5|) - |10|2
= 5(12 - 5) - 100
= 5(7) - 100
= 35 - 100
= -65
12. To order the numbers, first change the fractions to decimals and the percent to a decimal. Round to two
decimal places, if needed.
Then, compare the decimals.
Therefore, the numbers from least to greatest are shown below.
13. Factor using the difference of two squares.
x4 - 1 = (x2 - 1)(x2 + 1)
= (x - 1)(x + 1)(x2 + 1)
14. Find the greatest common factor (GCF) of the monomials by finding the GCF of the constants and of each
of the variables.
The GCF of 14 and 10 is 2.
The GCF of x3 and x3 is x3.
The GCF of y2 and y4 is y2.
The GCF of z3 and z0 (z0 = 1) is z0 = 1.
Therefore, the GCF of the two monomials is 2x3y2.
15. Since it is addition of two polynomials, drop the parenthesis and collect like terms.
(6x2 + 3x + 8) + (4x2 - 8x - 3) = 6x2 + 3x + 8 + 4x2 - 8x - 3
= (6x2 + 4x2) + (3x - 8x) + (8 - 3)
= 10x2 - 5x + 5
16. Find the least common multiple (LCM) of the monomials by finding the LCM of the constants and of each
of the variables.
The LCM of 9 and 5 is 45.
The LCM of u3 and u is u3.
The LCM of v4 and v4 is v4.
The LCM of w3 and w2 is w3.
Therefore, the LCM of the two monomials is 45u3v4w3.
17. To order the numbers, put each number into decimal form.
Next, order the decimal forms of the numbers.
Therefore, the numbers from greatest to least are as follows.
18. Find the greatest common factor (GCF) of the monomials by finding the GCF of the constants and of each
of the variables.
The GCF of 9 and 33 is 3.
The GCF of x and x is x.
The GCF of y2 and y is y.
Therefore, the GCF of the two monomials is 3xy.
19. If the bases are different but the exponents are the same, then multiply the bases and keep the exponent.
20. Even though 720 is not a perfect square,
another number.
can be simplified by factoring 720 into a perfect square and