math 1311 – business math i - angelo state university · web viewgiven two or more natural...

Math 1311 – Business Math I Day 1: Review Sets of Numbers

____ = { 1, 2, 3, 4, ..... } is called the set of __________________________. Let x be one of these numbers. Find the solution of the following equations.

2x + 4 = 6 x = _________ x + 2 = 2 x = ___________

_____ = { 0, 1, 2, 3, ... } is called the set of _________________________ Let x be one of these numbers ( ) . Find the solution of the following equations.

x – 4 = 6 x = ___________ x + 6 = 2 , x = ____________

______ = { ..., -3, -2, -1, 0, 1, 2, 3, ... } is called the set of __________________ Let x be one of these numbers. Find the solution of the following equations.

x – 4 = - 6 x = _____ 2x + 1 = 2 x = _________

Other Sets and other names for the sets above

With respect to integers the set { 1, 2, 3, ... } is also called the set of __________________________________ With respect to integers the set { 0, 1, 2, 3, 4, .... } is also called the _____________

{ 2, 3, 5, 7, 11, ... } represents the set of ____________________________ If a natural number is not prime and is greater than 1, then it is called _____________

The set that contains all real numbers that can be written as fractions is called the set of __________________________ We describe this set in what is called set-builder notation: Q = { a / b | a is an integer and b is a nonzero integer}

Real numbers that are not rational numbers are called _________________________________ ( this means the real numbers are made up entirely of rational and irrational numbers)

If a real number can not be written as a fraction – we call it an ______________________________

Rational: ________________________ Irrational:______________________________

Real numbers

Rational Irrational #’s

pure fractions integers

negative integers nonnegative integers (whole numbers)

zero positive integers (natural numbers, counting numbers)

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Properties of Real Numbers

commutative law of addition commutative law of multiplication a + b = _________ ab = ________

examples: ( 3 ) + ( - 4 ) = _________ ( 2/3 ) • ( 5/7) = ____________

associative law of addition associative law of multiplication a + ( b + c ) = _________ ( a b ) c = ____________

examples:

1.3 + ( 3.7 + 4.8) = _____________ ( • ) •

Distributive law of multiplication over addition a(b + c ) = ________________

examples: 3 • ( -2 + x ) = __________ 1.7 ( 2.9 – 1.2 ) = _______________

commutative and the associative laws do not hold true with subtraction and division – do they ?

ex. 2 – 4 = 4 - 2 ? ________ 12 6 = 6 12 ? ____________

ex. ( 12 – 4 ) - 2 = 12 - ( 4 – 2 ) ? _________ (16 8 ) 2 = 16 ( 8 2 ) ? ________

Other terms

Given a natural number(greater than 1): the number is either prime or it is composite. If it is composite, it can be written as a product of prime numbers ( prime factors). The product is called the prime factorization of the number.

ex. 24 = _____________ ex. 150 = ____________

ex. 95 = ____________ ex. 29 = ______________

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GCF - greatest common factor Given two or more natural numbers we can find the greatest ( largest) number that will divide evenly into all of the given numbers. This is called the greatest common factor – GCF ( or sometimes: the GCD – greatest common divisor)

Find GCF of

12 and 20 ___________ ( 8, 20, 30 ) = ___________

248 and 324

LCM – least common multiple Given two or more natural numbers we can find the smallest (least) number that all the given numbers will divide evenly into. Find LCM ( 20, 15 ) = ________ LCM ( 2, 3, 5 )

Find GCF and LCM of 150 and 240. _______________ ____________________

Absolute Value of a real number: Let x be a real number. The absolute value of x, written | x |, is defined by

x if x > 0 | x | = either - x if x < 0

ex. | 6 | = _________ | - 2 | = _________ - | - 4 | = _________,

| x | = _______ if x is a natural number

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Inequalities < : means less than : means less than or equal to ex. 4 < 6 we say 4 is less than 6

> : means greater than : means greater than or equal to ex. 12 10 we say 12 is greater than or equal to 10

True or False 4 < - 4 - 2 - 2 3/ 11 > 2 / 9 0.01 < 0.0091

______________ ______________ ______________ ________________Using inequalities we can represent numbers on a number line –

x < 4 x 2 - 4 < x 2

Integer Exponents

xn : exponential notation, x is called the ___________ and n is called the ____________ or the ___________

and xn = xxxx … x ( a total of n x’s) – this is the expanded notation

ex. 25 = _____________________ ( - 4) 3 = __________________ - ( 4)2 = _________

ex. – 3 3 3 3 = _____________ ex. – 42 = _____________ ( 2/3 )4 = ___________

ex. 0.014 = ________

Def. If x is a real number not equal to zero, then xo = ___________

( 4)0 = _______ ( - 2 ) 0 = _______ , ( - c) 0 = _________ for c 0 ( 0) 4 = ________

What about - ( 4 ) 0 = _________ So, - 3 0 = ___________ and - ( - 2 )0 = ______ . What is 0 0 = ______

Def. If n is a natural number, then x – n = ________ . This gives us a way to work with negative exponents in terms of

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natural numbers.

ex. Find

2-2 = ___________ 4-3 = ____________ 8-1 = __________

What about ( ¼) – 2 = __________ ( ½) – 3 = _____________ ( ¾)-2 = __________

Properties of exponents. Let x and y represent real numbers and let m and n be integers.

1. xn xm = ____________ x4 x5 = ____________ x8 x = ___________

2. xn xm = ______________ x 8 x4 = ___________ x12 / x20 = __________

3. (xn ) m = _______________ (x3 )2 = _________ (x6 )3 = _________

4. ( x y ) n = _____________ (2x)3 = _____________ ( 4xy)3 = ___________

( x2y3 )2 = _________ ( 2xy3 )3 = _____________

5. ( x y ) n = ____________ ( x/y)4 = __________ ( x2 / y3 ) 3 = ____________

Other examples.

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1) ( 2x3 y) ( - 4x2y4 ) = __________________ 2) ( 2xy2 )2 ( 3x2y )3 = _____________

4xy - 2y3

3) ----------- ------------- = _______________ 12xy6 x2y

-2x 3xy3

4) ( --------- )2 ( ---------- )2 = _____________ y2 -2y3

Since we have a definition for negative exponents – how would we work problems with negative exponents.

1. ( x - 2)- 4 = _______________ ( x-3y-4)-2 = _______________

( x-3 )2 = ______________ ( 2x-2 )-2 = ________________

2. ( 2x-3y0 ) ( 3x-1y ) = ________________

Since the rules work with integral exponents – do they work with rational exponents ? Yes

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1) x 2/3 x7/3 = ___________ 2) x 1/5 y x y 2/3 = ___________ 3) ( x -2/3 ) –6 = _____________

4) ( - 5x1/2y)1/2 = __________

5.

More Examples of exponent problems –Find

( - 2xy3 ) 0 = ____________ if neither x nor y = 0 180 = ___________ - 40 = _____________

0 4 = ____________ 0 - 2 = ____________ 00 = _________ - 42 = ___________ - 4 – 2 = _____________

Also, if x = -1 and y = - 2 and z = 0 find

1) xy ____________ 2) y0 = __________ 3) - x2 = ____________ 4) (x – y ) x + y = ___________

Radicals

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square roots: of 49 ==> ________________ square roots of 81 ==> ________________

Cube roots of 64 ==> ______________ cube roots of - 27 ==> ________________

Continue by asking for 4th roots, 5th roots,….Notice that each number has either 1, 2, or none nth roots.

ex. Find the square roots of 25 ______________ Find the fifth roots of – 32 ___________

Find the 4th roots of ( -16) ___________

n ___Define Principal nth roots of x: \/ x ; n is called the _____________, x is the _____________

n __ 4 ____ if x > 0, then \/ x > 0 \/ 125 = ____________

n __ 3 ____ if x < 0 and n is odd, then \/ x < 0 \/ - 27 = _________

n __ 2 ___ if x < 0 and n is even, then \/ x has no real value \/ -4 = ____________

ex. Find each of the following nth roots ___ 3 ___ 2 ____ 1) \/ 64 = _________ 2) \/ - 8 = ____________ 3 ) \/ - 25 = ____________

4 ___ 2 _____ 3 _____ 4) \/ x8 = ___________ 5) \/ x6y4 = ___________ 6) \/ 8x3y12 = __________

__ _____True or False: \/ x2 = x ___________________ So, \/ ( -2)2 = _________

___However, if we assume that x is a positive real number, then \/ x2 = x

n ___ Define x 1/n

= \/ x

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Examples of x1/ n

1) 8 1/3 = ____________________ 2) 16 ¼ = ___________________

3) - 25 ½ = __________________ 4) ( - 27 ) 1/3 = _______________

5) ( - 9 ) ½ = ________________ 6) ( 16) – 1 / 4 = _________________

n ___ n __ m

Define x m/n = \/ x m or ( \/ x )

Now we can use fractional exponents: 16 ¾ = __________, - 9 3/2 = ____________ , - 16-1/2 = ________

ex. (16x2/3y9 ) 1/ 4 = _____________ ex. ( -8x6y9 ) 2/3 = _______________

ex. ( 4x-4y8 ) – ¾ = _______________

More on radicals:

While we can simplify quantities that are perfect squares, perfect cubes,… 3

___ 4 _______ such as \/ -64 = _______ or \/ x8 y12 = __________

What about ___ 3 __ 3 ___ \/ 20 = _________ \/ 16 = ________ \/ x8

Properties:

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n ___ n __ n __ n __ n __ ____ 1) \/ xy \/ x \/ y = \/ x \/ y ex. \/ 4x2 = ______________

_______ n ___ ________ n / x \/ x / 9x4

2) \/ ----- = ----------- ex. / -------- = __________________ y \/ y \/ 4y2

ex. Write 42 in prime factored form ( as a product of prime numbers ) = ________________

ex. What is the prime factorization of 24 ? 24 = ______________

We say a radical is in simplest form if ( radical must be in prime factored form )

1) The index is smaller than all of the exponents inside the radical 5 ___ 4 ______ ex. \/ x8 = _________ ex. \/ 16x5y2 = ______________

2) There is no common factor between the index and all of the exponents inside the radical 4 ____ 6 ______ 4 ____ ex. \/ x2

= ________ \/ 8x3y3 = ____________ \/ 2x2

= __________

3) There can be no radical in the denominator or no denominator inside the radical.

ex. 2 4x -------- = __________ ex. --------- = ____________ \/ 2 \/ 4x

_______

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ex. / 9x / -------- \/ 4y

________ ex. / 9x 3 / -------- \/ 4y2

Other radical problems: sum/difference: __ 3 __ Find 4 \/ 9 = ________ 9 + \/ 8 = _______________

__ __ ___ 4 + 2 \/ 9 = _________ 2\/ 3 - 3 \/ 12 = _____________

products / quotients. __ ___ __ ___ \/ 5 \/ 20 = ___________ \/ 4x \/ 2x2 = ___________

______ __ 3 __ 3 / ___ \/ 2 \/ 2 = ______________ \/ \/ 8 = _______________

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Polynomials (factors, terms, degree)

Sum of literal expressions in which each term consists of a product of constants and variables with the restriction that each variable must have a nonnegative integer exponent.

2x, 3x2y – 4, 1 + x + 3x2, 5 – 3xy + y9, …. What about __________ or ____________ ?

Special types of polynomials if one term: __________ if two terms: ___________ if three terms: __________

How many terms does each of the polynomials have ?

2xy 3 + 2xy + x9 + y2 1 + x + x2 _______________ __________________________ __________________

Def. (degree) monomials : 3x ==> _________ 2x5 ==> ___________ ½ x5 y3 ==> ____________

binomials: 2x – 1 ==> ________ x + y ==> _________ 3x2y - 4x3y2 ==> ____________

other polynomials: x – 2xy + y3 ==> ___________ x10 - 2xy + x6y5 ==> _________-

Basic operations of polynomials: sum – difference, products - quotients Find ( sum and differences)

a) (2xy - 4x ) + ( 3xy - 2x ) ==> _____________

b) ( 3x2 - 2x + 3 ) - ( 2x2 - 4x - 5 ) ==> __________

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Products

a) 2(x – 3y ) = _______________ b) x( x + 2y ) = ___________________

c) 3x3 ( 3xy – 2x4y ) = ______________

d) ( 2x –3y )( 4x + 2y ) = _________________

Def. The process of writing a polynomial as a product of other polynomials of equal or lesser degree is called

__________________

Recall: GCF

Find GCF ( 20, 36 ) = ____________ GCF ( xy, x ) = __________ GCF(2xy, 6y2 ) = _________

GCF ( 12x3y6, 8x4y2 ) = ___________

Special Products and Factoring

1) greatest common factor: x( y + x) = ________________

2) difference of squares : (x – y ) ( x + y ) = _____________

3) sum – difference of cubes : ( x – y ) ( x2 + xy + y2 ) = ___________

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4) perfect squares: (x + y )2 = __________________ = ________________

5) Trinomials of the form ax2 + bx + c

Review of methods of factoring - Factoring: process of writing a polynomial as a product of other polynomials of equal or lesser degree.

Methods: GCF – always look for a common factor - 1st method

Difference of squares: x2 - y2 = (x – y ) ( x + y)

Sum – Difference of Cubes: x3 + y3 = ( x + y ) ( x2 - xy + y2 ) , x3 - y3 = ( x – y ) ( x2 + xy + y2 ) -- SOPPS

Perfect Squares: x2 + 2xy + y2, first and last must always be positive

Multiply (x + 2y )2 = _____________ (3x – 4y )2 = _____________________

Factor: x2 - 8x + 64 = _______________ x2 + 10x + 25 = __________________

x2 + 4xy + 4y2 = ___________ 4x2 - 12xy + 9y2 = _________________

x2 - 16x - 16 = ___________

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Grouping –

2 ( x – y ) - y ( x – y ) = _________________ xy + 2x - y2 - 2y = _______________

x2 - y2 – 2y - 4 = ______________________ x3 - y3 - x + y = _______________

More on Factoring:

GCF: 2 – 12x = ___________________ x(y-1) – y( y – 1 ) = ________________

x2 + 2x = ___________________ 2x ( y - 4 ) + 3y( 4 – y ) = _______________

Difference of Squares:

x2 – 36 = _________________ x3 - 4x = _____________________

12x2y - 27y3 = _______________ x4 - 16 = ____________________ x2 + 9 = ________

Perfect Squares:

x2 + 12x + 36 = _____________________ x2 – 8x + 64 = __________________

x2 + 10x + 25 = _____________________ x2 – 4x - 4 = ____________ x3 – 6x2 + 9x

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Sum/Difference of Cubes:

x3 - 8 = ________________________ x3 + 27 = ________________________

8x3 + 64 = ___________________________ 1 - 8y3 = __________________________

Trinomials of the form ax2 + bx + c

Case I : a = 1, factor x2 + bx + c

x2 + 6x + 5 = ______________________ x2 + 8x + 7 = ________________

x2 - 8x + 15 = ____________________ x2 - 6x + 8 = ___________________

Case II: If a 0, factor ax2 + bx + c

2x2 + 5x + 2 = _______________________ 3x2 - 8x + 4 = ____________________

4x2 - 4x - 3 = ________________________ 4x2 + 2x - 6 = ____________________

Grouping: 2xy – 3x - 4y + 6 = _________________

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Algebraic Fractions -- quotients of polynomials.

ex. 2x / ( x2 - 3x + 4 ) , ( x - 4 ) / 3, ( x2 + 1 ) / (x2 + 2x + 1), ¾, …

We can reduce these fractions to equivalent fractions with a smaller denominator and we can perform algebraic operations on these fractions: add, subtract, multiply, divide, ….

ex. Reduce each of the following fractions to lowest terms x – 1 2x + 1 a) -------- = b) ------------- x2 - 1 8x2 + 4x

x2 + 2x - 8 c) ----------------- x2 + 4x

Multiply / Divide:

x2 + x 2x - 2 a) ----------- ------------ x2 - 1 4x + 8

x3 + 1 x2 + x b) ------------ ----------------- x2 - x + 1 2x

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x2 - 2x - 3 x2 - 4x + 3 c) ---------------- ------------------ x2 - x - 2 3x2 - 3

Add/Subtract 2x – 4 4x – 3 a) --------- + ------------- = x2 + 5 x2 + 5

3x - 4 2x - 7 b) ----------- - -------- 2x + 3 2x + 3

x – 4 2x - 1 c) ----------- - ---------- x + 3 x + 3

Other Examples -

x - 1 21) ---------- + -------------- 4x x

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2x + 1 32) ---------- - -------- 2x2 x

x + 1 x - 13) --------- - ------------ x - 1 x + 2

3 + x x + 24) ---------- - -------- x + 1 x 2 - 1

2 2 - 2x 5) -------- + ---------- x - 2 2 - x

2/x - ½ 6) --------------- 3/x + ½

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1 / ( x – 1 ) 7) --------------------- 3 - 1/( x – 1 )

18. 1 - --------------------------- 1

1 - ------------ 1 - 1/x

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Solving Equations –

x + 2x = x + 4: solution set ____________ x – 2 = 0 solution set _______________ These two equations have the same solution.

two equations are said to be equivalent provided they have the same solution set.

Reduce equations to an equivalent form whose solution is easy to read – we may use the following two operations

1) You can add (subtract) any quantity to both sides of an equation to obtain an equivalent equation

x + 4 = 3 _____________ x – 5 = - 3 ____________

2) You can multiply (divide) both sides of an equation by any nonzero value

2x = ½ ____________ x/4 = - 2/3 _____________

Other Examples -

1) 2x - 4 = 3 _____________ 2) x/4 + 6 = 1 _____________

3) 3n/5 - n = - 2 ____________ 4) 2 - 3( 1 – 2x ) = 1 ____________

5) 3/x + ½ = 4/x _____________ 6) ( x – 1 ) / 2 - ( x + 1 ) / 3 = 1 __________

2 1 1 7) ------- + ------ = ------- x + 1 3 x + 1

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8 ) another example

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Quadratic equations –

An equation of the form ax2 + bx + c = 0 is called a quadratic equation in x

ex. x2 = 5x – 2 a = _____ b = _________ c = __________

ex. 1 – x2 = 2x a = ______ b = _________ c = __________

To find the solution of equations of this form we may use one of the following methods:

1) factoring: either ax2 + bx = 0, or ax2 + bx + c = 0 2) square root method: ax2 + c = 0 3) completing the square: any equation of the form ax2 + bx + c = 0 4) Quadratic formula: any equation of the form ax2 + bx + c = 0

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Short - Quiz #1 Name ___________________________________________ Math 1302 - – January 17, 2002

1. Label each of the following sets by using

Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers, Real Numbers.

a) { 0, 1, 2, 3, … } _____________________________________________

b) { a/b : a is an integer and b is a nonzero integer } _______________________________

c) the set is made up entirely of all rational and all irrational numbers _______________________

2. Find the smallest natural number. ________________

3. What is the smallest positive integer ? _________________ __4. What kind of real number is \/ 5 ? A real that is also a _______________________________________

5. Which of these examples illustrates the commutative law of addition ? ___________________

3 + 4 = 4 + 3 ( 3 + 4 )+ 5 = 3 + ( 4 + 5 ) 3 ( 4 + 5 ) = 3(4) + 3 (5)

6. Which of the following examples illustrates the associative law of multiplication ? ___________________

3(4) = 4(3) (34) 5 = 3(4 5 ) 3(4 + 5 ) = 3(4) + 4(5)

7. Find the absolute value of

a) | - 4 | = ________________ b. - | - 2 | = _____________

8. How many whole number solutions does the equation x2 = 16 have ? ______________________

__ How many rational solutions does the equation x + \/ 3 = 0 have ? ________________

9. Find GCF ( 12, 28 ) = __________________ LCM ( 20, 24 ) = _______________

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Math 1302 ----------Week 2 – Spring Semester 2002

Other Radicals:

a)

b)

c)

d)

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Name ___________________________________ Math 1302 Jan. 22, 2002 LongQZ # 1

Note: to grader --- each blank is worth ½ point --- adds up to 16 points – start off everybody with 4 points for a total of 20 points.

1. Use the terms that follow to identify each of the sets set of natural numbers, set of whole numbers, set of integers, set of rational numbers, set of irrational numbers, set of real numbers, set of prime numbers

{ 1, 2, 3, 4, 5, ... } is the set of _____________________

Natural numbers that are divisible only by 1 and themselves and are greater than 1 are called ___________________ The set { ........... – 3, - 2, - 1, 0, 1, 2, 3, ... } is called the set of ___________________

Another name for the set of positive integers is the set of _______________________

All real numbers that can not be written as fractions make up the set of _____________________

The real numbers are made up entirely of what two sets? ___________________ and ___________________

2. Identify which law(property) is being used. ( write the complete description; commutative law of addition) commutative law of addition, commutative law of multiplication, associative law of addition, associative law of multiplication , distributive law , or None of these

2( x + y ) = 2x + 2y ==> _____________________________ x(y) = y(x) ==> _____________________________ x + (y + z ) = (x + y) + z ==> __________________________

12 – 4 = 4 - 12 ==> ________________________

3. Find each of the following absolute values – exact values – no calculators needed.

- | - 6 | = _________ | 2 + | = ______________

4. Find each of the following

a. GCF ( 12, 8 ) = ___________ b. GCF ( 20 , 15 ) = _________ c. LCM ( 8, 6 ) = ___________

5. Simplify each of the following exponent x4 x12

a. x4 x8 = _________ b. ( x4 ) 5 = _________ c. ---------- = _________ d. -------- = _________ x12 x8

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e. ( -2x2y )2 = ___________

6. Simplify. Assume that x 0.

a) - 4 2 = _______________ b) 03 = __________________

c) ( 4x3 )0 = ____________ d) 4 - 2 = ________

7. Simplify. Leave answers with only nonnegative exponents.

a) (-2x3y - 3 )2 = _____________ b) ( -2xy - 2 )( - 3x2 y2 ) = ___________

c) ( ¾ ) – 1 = ___________

8. More exponents. 4x1/2

a) x1/2 2x3/2 = __________ b) --------------- = ____________8x3/2

c) ( 41/2 )3/2 = __________

9. One last exponents.

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Name _________________________________ Math 1302 Quiz, January 24, 2002 ---- Short Quiz ( 10 points)

1. Find the square roots of 100. __________________ Find the cube root of -27 . _____________

2. Find __ 4 ___ a) \/ 81 = ___________________ b) \/ 16 = _____________

_______ ____ c) \/ 25 49 = ______________ d) \/ - 4 = ____________

3. Find 4 –2 = ______________ ( ¼ ) – 1 = _________________

4. If x = -2 and y = 0 , then find

x2 - xy = ___________

5. Find x1/3 2x1/2 = _________________

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Name ____________________________________ Math 1302 – January 29, 2002 -- Long Quiz 20 points (Note: count each blank 1 point except for #9. Count #9 as three points )

1. State the five rules of exponents –

a) xm xn = ___________ b) xm xn = ______________ c) ( xm )n = ___________

d) ( x y )n = _______________ e) ( x / y ) n = _______________

2. Simplify using the rules of exponents. All exponents should be positive. Write in simplest form with no radicals.

a) x1/2 x1/3 = ____________ b) ( 4x6 ) ½ = ______________

c) ( - 8x1/3y- 6 )-3 = __________________ d) ( - 8/27 )-1/3 = _____________

3. Write each of the following radicals in simplest form . 3 ________ ___ a) \/ -8x6y12

= ___________________ b) \/ 40 = ___________

__________ c) \/ 900 + 1600 = _______ d) ( - 64x9y - 12)-1/ 3 = ____________

4. More radicals. Write in simplest form.

8 ___ 4 ______ a) \/ x4 = ____________ b) \/ 4x2y4

= ___________

_____ __ __ c) \/ 8x3y5 = ___________ d) \/ 8 + \/ 50 = __________

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9.

Name _______________________ Math 1302 – January 31, 2002 --- SHORT Quiz – 10 points

1. A polynomial with two terms is called a ___________________

2. How many terms does the following polynomial have ? _____________________

3. What is the degree of each of the following polynomials

a) 2 – x ___________

b) 4 + 2x3 - 6x + y _________

c) 3 _____________

4. Which of these is not a monomial ? ____________________

2, - 4x , x2y , 1 - 2x , all are

5. Add the following polynomials.

( 3x2 + 4x - 2 ) + ( 2x2 – 4 ) = _________________________________

6. Find the difference of the following polynomials.

( 2 – x – x2 ) - ( 4 + 5x – 2x2 ) = _________________

7. What is the product of

a) 2 ( x2 - 4x + 4 ) = _________________

b) 2xy2 ( 3 – 2x + xy ) = ________________________

c) ( x + 2y)(4x + 3y ) = ____________________

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d) ( x + 2y )2 = ____________________________

Name __________________________________ Math 1302 - February 5, 2002 --- LONG Quiz – 20 points

0. 1 Simplify to a single number.

40 = ____________ -22 = _______________ 4 / 0 = ____________

1. Use the rules of exponents to simplify.

a) 2x3y 4xy4 = ________________ ( -2x2y4 )3 = ______________________

4x-2 y3 c) ( --------------- )2 = ____________________ -2x3y -2

2. Use the rules of radicals to simplify 3 _____ 4 ____ a) \/ 8x6y12 = __________________ b) \/ 4x2 = ____________

3. More radicals 4 a) -------- = c) ----------------- \/ 2

4. What is the degree of the following polynomials.

a) 210 - x _______________ b) x4 + y3x2 _____________

5. perform the given operation and reduce to lowest terms (simplest ) .

a) 4x(x2 - 2x ) = ____________________ b) ( x – 2y)(x + 2y ) = ____________________

6. Find GCF( 12, 40 ) = _______________ GCF ( 4x2y, 10xy3 ) = ______________

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7. Factor each of the following polynomials.

4x – x = _________________ 8xy – 2y2 = _________________ __________ x2 - y2 + 2y - 1 = ___________________________

Name ________________________________ Math 1302 - February 7, 2002 ---- Short Quiz ( 10 points )

Factor each of the following polynomials.

1) 2x + x = ___________________ 2) x ( y – x ) - y ( y – x ) = _____________________

3) x3 - 4x = ______________________________

4) x3 + 64 = _____________________________________

5) x4 - 16 = ________________________________________

6) x2 + 6x + 5 = ___________________________________

7) x2 + 12x + 36 = __________________________________

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8) x2 + 4 = _________________________________________

More examples:

1) 8/89. Sally took four tests in science class . On each successive test, her score improved by 3 points. If her mean score was 69.5 %, what did she score on the first test ?

2) 12/89 A college student earns $20 per day delivering advertising brochures door-to-door, plus 75 cents for each person he interviews. How many people did he interview on a day when he earned $56.

3) 16/89 A builder wants to install a triangular window with the angles shown below. What angles will he have to cut to make the window fit ?

Other example:

x - 1 x + 1 ------ - ---------- = 1 2 3

Test I review:

Know sets of numbers – natural #’s, whole #’s, integers, rational, irrational, real, prime, nonnegative integers, positive integers be able to classify a number as a member of a particular set be able to give an example of a number that is a member of one set but not another compare sets, know how they differ in terms of the elements (numbers) in the set

Properties of sets –

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commutative law of addition, commutative law of multiplication, associative law of addition, associative law of multiplication , distributive law

Properties and definitions of numbers absolute value , GCF, LCM inequalities on a number line order of operations: PEMDAS scientific notation evaluating expressions

basic operations be able to add, subtract, multiply, divide any real number

exponents know the five basic properties work with positive integer exponents , exponent of zero, negative coefficients or signs , negative exponents, fractional exponent

radicals – be able to simplify and work problems as done in class and over homework , fractional exponents and radicals

polynomials – basic operations - add, subtract, multiply, divide, degree, terms, factors, classify – monomials, binomials, trinomials, long-hand division

factor polynomials; GCF, difference of squares, perfect squares, sum of cubes, difference of cubes, trinomials, grouping make sure to work with any combination of the above methods

algebraic fractions: add, subtract, multiply, divide, complex fractions

linear equations in one variable - solve, find solution set solve for a variable in a formula word problems

Identify equations as linear, quadratic equations state methods of solutions for quadratic equations Write a quadratic equation in its standard form, identify a, b, c Write down quadratic formula

Name _______________________________ Math 1302 – February 12, 2002 Long Quiz 20points

1. Factor each of the following polynomials.

a) x2 – 16y4 = ________________________

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b) 2x2 - 5x + 4 = ___________________

c) 8x4 - xy3 = ________________________

2. More factoring

a) 4( x – 2y ) + 2y ( x – 2y ) = _________________________

a) 2x – xy + 6y - 3y2 = ___________________________

3. Reduce each of the following fractions .2x

a) ------------ = 2x2 - x

x3 + 8 b) ----------------- = x2 - 2x + 4

HW: p. 24: 54, 57, 60, 63, 66, 69, 72, 82, 91, 94, 97, 119, 122, page 39: 11, 14, 17, 20,

Name ___________________________________ Math 1302.F10 Jan. 17, 2001 Quiz # 1

1. Use the terms that follow to identify each of the sets

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set of natural numbers, set of whole numbers, set of integers, set of rational numbers, set of irrational numbers, set of real numbers, set of prime numbers

{ 0, 1, 2, 3, 4, 5, ... } is the set of _____________________




The set of _______________ is made up of only rational and irrational numbers.

2. Identify which law(property) is being used. commutative law of addition, commutative law of multiplication, associative law of addition, associative law of multiplication , distributive law , or None of these

2( x + y ) = 2x + 2y ==> _____________________________ x(y) = y(x) ==> _____________________________ x + (y + z ) = (x + y) + z ==> __________________________

12 – 4 = 4 - 8 ==> ________________________

3. Find each of the following absolute values

| 9 | = ____________ | - 2/ 3 | = ______________

- | - 6 | = _________ | 2 + | = ______________


a. GCF ( 12, 8 ) = ___________ b. GCF ( 20 , 15 ) = ______________

c. LCM ( 8, 6 ) = ___________


a. x4 x8 = _________ b. ( x4 ) 5 = _________ c. ---------- = _________ d. -------- = _________ x12 x8

Name ___________________________________ Math 1302.F10 Jan. 17, 2001 Quiz # 2 HW: p. 24: 54, 57, 60, 63, 66, 69, 72, 82, 91, 94, 97, 119, 122, page 39: 11, 14, 17, 20,


set of natural numbers, set of whole numbers, set of integers, set of rational numbers, set of irrational numbers,

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set of real numbers, set of prime numbers

{ 1, 2, 3, 4, 5, ... } is the set of _____________________

The set { ........... – 3, - 2, - 1, 0, 1, 2, 3, ... } is called the set of ___________________

All real numbers that can be written as fractions make up the set of _____________________


2( x + y ) = 2x + 2y ==> _____________________________

x + (y + z ) = (x + y) + z ==> __________________________


a. | 2 - | = ____________ b. - | - 2/3 | = ___________


a. GCF ( 12, 24 ) = ___________ b. LCM ( 12 , 24 ) = ______________

5. Simplify each of the following exponent 8x4

a. ( - 2x4 ) (3 x8 y) = ____________ b. ( 2x4 ) 5 = _________ c. ---------- = _______ 6x12

6. Find

a. b. c. d.

e. f. g.

W: p. 12: 1 – 20 all (except for 15-17), 21, 26, 31, 36, 41, 46 51, 56, 61, 66, 67, 71, p 24: 1-12 all, 15, 33, 36, 39, 42, 45, 48, 51, HW: p. 24: 54, 57, 60, 63, 66, 69, 72, 82, 91, 94, 97, 119, 122, page 39: 11, 14, 17, 20,

Name ___________________________________ Math 1302.F10 May 31, 2001 Quiz # 2


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{ 0, 1, 2, 3, 4, 5, ... } is the set of _____________________

The set that contains the whole numbers and their opposites is called is called the set of ___________________



xy = yx _____________________________ x (y z ) = (x y) z ==> __________________________


a. | 4 - | = ____________ b. | x | = ___________ if x is a negative integer


GCF ( 30, 24 ) = ___________

5. Simplify each of the following exponent 8x4

a. ( - 2x4 ) (3 x8 y) = ____________ b. ( 2x4 ) 5 = _________ c. ---------- = _______ 6x12

6. Find

a. – 4 - 2 = _________ b. – 16 –1/2 = _________ c. – 8 2/3 = ________

e. ( - 16 ) ½ = ________

7. Simplify.

a. ( 2x3y-2)-3 = __________ b. \/ 64 = _________ c. 2 \/ 25 - 4 \/ 36 Name ___________________________________ Math 1302.F10 Jan. 17, 2001 Quiz # 4


{ 1, 2, 3, 4, 5, ... } is the set of _____________________

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The set that contains the whole numbers and their opposites is called is called the set of ___________________

All positive integers _____________________

2. Identify which law(property) is being used. x+y = y+x _____________________________ x (y z ) = (x y) z ==> __________________________


a. | - 4 - | = ____________ b. | x | = ___________ if x is a whole number


LCM ( 30, 24 ) = ___________

5. Simplify each of the following exponent 8x4y-2

a. ( - 2x-4 ) (3 x-8 y) = ____________ b. ( 2x44 ) -5 = _________ c. ---------- = _______ 6x-12 y-4

6. Find

a. ( – 4 ) - 2 = _________ b. ( – 16 –1/2) = _________ c. – 8 -2/3 = ________

7. Simplify.

a. b. c.

8. what is the degree of the following polynomials ?

a. b. c.

Math 1302 – Quiz # 3

1. Use the numbers; 0, - 2, 3, ½, 0.01001000100001... to answer the questions that follow.

Give me an example of

a) an integer that is not a whole number ______________ b) a rational number that is not an integer _______

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2. Which property is being use; ( a + b) = ( b + a ) _______________________________________________


a) - 4 2 = _____________ b) 16 -1 / 2 = ______________ c) 0 4 = _______________

d) - 40 = ____________ e) - 251/2 = ______________ f) 0 / 4 = ______________

g) - 163/4 = ___________ h) ( - 8/ 27 ) 2/3 = _______________

4. Use the rules of radicals to simplify.

a) 16x8y12 = ______________ b) \/ 8x3 = _____________

4 c) \/ 8x3y7 = ______________ d) \/ --------- 9x

e) 3 \/ 16 - 4 \/ 4 = _______ f) 3 \/ 50 = _________

4 g) ---------- \/ 8


a) ( - 8x6 y – 9 ) - 1/3 = ______________

16x – 1 y2

b) ( ------------------- ) 2

12 x -2 y – 2

c)

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Rational Exponents

More on RadicalsHW page 15: 1 – 15 all, 25, 28, 33, 47, page 20: 1 – 16, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 46a, 46b,

Name ___________________________________ Math 1311.030 Jan. 22, 2001 Quiz # 1



{ 0, 1, 2, 3, 4, 5, ... } is the set of _____________________




The set of _______________ is made up of only rational and irrational numbers.


2( x + y ) = 2x + 2y ==> _____________________________ x(y) = y(x) ==> _____________________________ x + (y + z ) = (x + y) + z ==> __________________________

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12 – 4 = 4 - 8 ==> ________________________


| 9 | = ____________ | - 2/ 3 | = ______________

- | - 6 | = _________ | 2 + | = ______________


a. GCF ( 12, 8 ) = ___________ b. GCF ( 20 , 15 ) = ______________

c. LCM ( 8, 6 ) = ___________


a. x4 x8 = _________ b. ( x4 ) 5 = _________ c. ---------- = _________ d. -------- = _________ x12 x8

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math 1311 – business math i - angelo state university · web viewgiven two or more natural...

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