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Basic Properties of Real Numbers: Proficiency Exam

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

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Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret. This module contains the proficiency for the chapter "Basic Properties of Real Numbers".

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Proficiency Exam

For the following problems, simplify each of the expressions.

Exercise 1

((Reference)) 8(6−3)−5⋅4+3(8)(2)÷4⋅3 8(6−3)−5⋅4+3(8)(2)÷4⋅3

Solution

40

Exercise 2

((Reference)) { 2 (1+7) 2 } 0 { 2 (1+7) 2 } 0

Solution

1

Exercise 3

((Reference)) 1 8 + 4 0 + 3 3 (1+4) 2 2 (2+15) 1 8 + 4 0 + 3 3 (1+4) 2 2 (2+15)

Solution

137 68 137 68

Exercise 4

((Reference)) 2⋅ 3 4 − 10 2 4−3 + 5( 2 2 + 3 2 ) 11−6 2⋅ 3 4 − 10 2 4−3 + 5( 2 2 + 3 2 ) 11−6

Solution

75

Exercise 5

((Reference)) Write the appropriate relation symbol (>,<) (>,<) in place of the * * .
 5(2+11)∗2(8−3)−2 5(2+11)∗2(8−3)−2

Solution

> >

For the following problems, use algebraic notation.

Exercise 6

((Reference)) (x−1) (x−1) times (3x plus 2) (3x plus 2) .

Solution

( x−1 )( 3x+2 ) ( x−1 )( 3x+2 )

Exercise 7

((Reference)) A number divided by twelve is less than or equal to the same number plus four.

Solution

x 12 ≤( x+4 ) x 12 ≤( x+4 )

Exercise 8

((Reference)) Locate the approximate position of −1.6 −1.6 on the number line.

A real number line with arrow on each end, labeled from negative three to three in increments of one.

Solution

A number line with arrows on each end, labeled from negative five to five in increments of one. There is a closed circle at a point between negative two and negative one, labeled as negative one point six.

Exercise 9

((Reference)) Is 0 a positive number, a negative number, neither, or both?

Solution

Zero is neither positive nor negative.

Exercise 10

((Reference)) Draw a portion of the number line and place points at all even integers strictly between 14 and 20.

Solution

A number line with arrows on each end, labeled from fourteen to twenty in increments of one. There are closed circles at sixteen, and eighteen.

Exercise 11

((Reference)) Draw a portion of the number line and place points at all real numbers strictly greater than −1 −1 but less than or equal to 4.

Solution

A number line with arrows on each end, labeled from negative four to five in increments of two. There is a closed circle at four, and an open circle at negative one. These circles are connected by a black line.

Exercise 12

((Reference)) What whole numbers can replace x x so that the following statement is true? −4≤x≤5 −4≤x≤5 .

Solution

0, 1, 2, 3, 4, 5 0, 1, 2, 3, 4, 5

Exercise 13

((Reference)) Is there a largest real number between and including 6 and 10? If so, what is it?

Solution

yes; 10

Exercise 14

((Reference)) Use the commutative property of multiplication to write m(a+3) m(a+3) in an equivalent form.

Solution

( a+3 )m ( a+3 )m

Exercise 15

((Reference)) Use the commutative properties to simplify 3a4b8cd 3a4b8cd .

Solution

96abcd 96abcd

Exercise 16

((Reference)) Use the commutative properties to simplify 4(x−9)2y(x−9)3y 4(x−9)2y(x−9)3y .

Solution

24 y 2 ( x−9 ) 2 24 y 2 ( x−9 ) 2

Exercise 17

((Reference)) Simplify 4 squared times x x cubed times y y to the fifth.

Solution

16 x 3 y 5 16 x 3 y 5

Exercise 18

((Reference)) Simplify (3)(3)(3)aabbbbabba(3)a (3)(3)(3)aabbbbabba(3)a .

Solution

81 a 5 b 6 81 a 5 b 6

For the following problems, use the rules of exponents to simplify each of the expressions.

Exercise 19

((Reference), (Reference)) (3a b 2 ) 2 (2 a 3 b) 3 (3a b 2 ) 2 (2 a 3 b) 3

Solution

72 a 11 b 7 72 a 11 b 7

Exercise 20

((Reference), (Reference)) x 10 y 12 x 2 y 5 x 10 y 12 x 2 y 5

Solution

x 8 y 7 x 8 y 7

Exercise 21

((Reference), (Reference)) 52 x 7 y 10 (y− x 4 ) 12 (y+x) 5 4 y 6 (y− x 4 ) 10 (y+x) 52 x 7 y 10 (y− x 4 ) 12 (y+x) 5 4 y 6 (y− x 4 ) 10 (y+x)

Solution

13 x 7 y 4 ( y− x 4 ) 2 ( y+x ) 4 13 x 7 y 4 ( y− x 4 ) 2 ( y+x ) 4

Exercise 22

((Reference), (Reference)) ( x n y 3m z 2p ) 4 ( x n y 3m z 2p ) 4

Solution

x 4n y 12m z 8p x 4n y 12m z 8p

Exercise 23

(5x+4) 0 (3 x 2 −1) 0 (5x+4) 0 (3 x 2 −1) 0

Solution

1

Exercise 24

x ∇ x □ y Δ x Δ y ∇ x ∇ x □ y Δ x Δ y ∇

Solution

The product of x to the power 'delta plus square minus triangle' and y to the power 'triangle minus delta'.

Exercise 25

((Reference), (Reference)) What word is used to describe the letter or symbol that represents an unspecified member of a particular collection of two or more numbers that are clearly defined?

Solution

a variable

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