Download Basic Algebra Practice Test 1. Exponents and integers: Problem

Basic Algebra Practice Test
1.
Exponents and integers: Problem type 2
Evaluate.
This is (9)( 9) = 81
This is (4)(4) = 16
2.
Exponents and signed fractions
Evaluate. Write your answers as fractions.
3.
Exponents and order of operations
Evaluate.
74
4.
Evaluation of a linear expression in two variables
Evaluate the expression when = 3 and = 6.
 +2
 (3) + 2(6)
 (3)  12
3  12
15
5.
Distributive property: Advanced
Use the distributive property to remove the parentheses.
6.
Combining like terms in a quadratic expression
Simplify the following expression.
and
7.
are the only like terms
Writing a mathematical expression
Write an algebraic expression to answer the question below.
If one notebook costs dollars, what is the cost, in dollars, of 8 notebooks?
The cost of 8 notebooks will be 8 times the cost of one notebook. Since one notebook
costs dollars, this will be 8 .
8.
Solving a two-step equation with integers
Solve for .
7 + 4 = 23
Simplify your answer as much as possible.
7 + 4 = 23
+7
+7
4 = 16
9.
Solving a two-step equation with signed fractions
Solve for .
Simplify your answer as much as possible.
Multiply both sides by the common denominator 12.
10. Solving a linear equation with several occurrences of the variable: Problem type 2
Solve for .
Simplify your answer as much as possible.
Multiply both sides by the common denominator 10.
11. Solving a linear equation with several occurrences of the variable: Problem type 3
Solve for .
Simplify your answer as much as possible.
Distribute
12. Solving a linear inequality: Problem type 3
Solve the inequality for .
Simplify your answer as much as possible.
Reverse the inequality sign when dividing by a negative number.
13. Graphing a linear inequality on the number line
Graph the inequality below on the number line.
The inequality sign is a , so use a filled in circle at .
shade to the left of 3.
5 4 3 2 1 0 1 2 3 4
is any number less than 3, so
5
14. Algebraic symbol manipulation: Problem type 2
Solve the following equation for .
Whenever an equation has a fraction, multiply both sides by the denominator.
15. Solving a fraction word problem using a simple linear equation
The yearbook club had a meeting. The meeting had 24 people, which is two-thirds of
the club. How many people are in the club?
This is the same as
24 people is two-thirds of the club
The word is will translate as an equal sign and of will translate as multiplication.
You can use for people.
Multiply both sides by the denominator
There are 36 people in the club.
16. Solving a word problem using a linear equation: Problem type 2
Customers of a phone company can choose between two service plans for long distance
calls. The first plan has no monthly fee but charges $0.14 for each minute of calls. The
second plan has a $15 monthly fee and charges an additional $0.09 for each minute of
calls. For how many minutes of calls will the costs of the two plans be equal?
We want:
Cost of first plan = Cost of second plan
Cost of first plan = $0.14 for each minute times the number of minutes
Cost of second plan = $15 monthly fee + $0.09 for each minute times the number of
minutes
You can use for minutes.
.
.
.
.
.
.
.
.
.
.
The costs of the two plans are equal for 300 minutes.
17. Set builder and interval notation
Graph the set
on the number line.
Then, write the set using interval notation.
is between the endpoints 3 and 2. The endpoint  has a sign, the inequality sign
there has an equal, so use a filled in circle at this endpoint. The endpoint 2 has a <
sign, so use an open circle. Shade between these.
•
5 4 3 2 1 0 1 2 3 4
5
The left endpoint of the interval is 3 and the inequality sign there has an equal, so use
a bracket for the left of the interval notation. The right endpoint of the interval is 2 and
the inequality sign there does not have an equal, so use a parenthesis there.
[3, 2)
18. Function tables
The function is defined by the following rule.
Complete the function table.
( )
5
14
1
2
0
1
2
7
3
10
19. Domain and range from ordered pairs
Suppose that the relation is defined as follows.
,
,
,
,
,
Give the domain and range of .
Write your answers using set notation.
The domain is the set of first values.
Domain: {3, 1, 6}
The range is the set of second values listed only once.
Range: { , 1}
20. Graphing a line given the x- and y-intercepts
Graph the line whose -intercept is 3 and whose -intercept is 2.
y
-intercept
x
-intercept
21. Graphing a line given its equation in slope-intercept form
Graph the line.
y
You can make a table of values for
points to plot.
0
1
2
x
3
1
1
22. Graphing a line through a given point with a given slope
Graph the line with slope 2 passing through the point (1, 3).
y
The slope is 2 =
x
1
2
.
To plot another point go either 2 in the
-direction and 1 in the -direction or go
2 in the -direction and 1 in the direction from (1, 3).
23. Graphing a vertical or horizontal line
Graph the line = 1.
y
x
The graph of = a number will
always be a vertical line that crosses
the -axis at the number.
24. Finding x- and y-intercepts of a line given the equation in standard form
Find the -intercept and -intercept of the line given by the equation.
4 +9 =8
For -intercept, let = 0 and solve for .
4 + 9(0) = 8
4 =8
=2
-intercept = 2
For -intercept, let = 0 and solve for .
4(0) + 9 = 8
9 =8
-intercept =
25. Finding slope given two points on the line
Find the slope of the line passing through the points (3, 4) and (8, 3).
slo e
slo e
26. Writing an equation of a line given the y-intercept and a point
Write an equation of the line below.
y
slo e
-intercept = = 2
1
2
x
The equation of a line is =
+
27. Writing the equations of vertical and horizontal lines through a given point
Write equations for the vertical and horizontal lines passing through the point (9, 3).
Vertical line: = 9
Horizontal line:
= 3
28. Evaluating expressions with exponents of zero
Evaluate the expressions.
Anything to the zero power is 1
1
29. Writing a negative number without a negative exponent
Rewrite the following without an exponent.
A negative exponent means reciprocal. This is the same as moving between the
numerator and denominator and changing the sign of the exponent.
30. Multiplying monomials
Multiply.
Simplify your answer as much as possible.
This can be rewritten as
When multiplying the same base, add the exponents.
31. Product rule of exponents in a multivariate monomial
Simplify.
Use only positive exponents in your answer.
This can be rewritten as
When multiplying the same base, add the exponents.
Move the variables that have a negative exponent to the denominator.
32. Quotients of expressions involving exponents
Simplify.
There is one factor in the numerator and 6 factors in the denominator. The one in
the numerator cancels with one in the denominator leaving 5 in the denominator.
There are 5 factors in the numerator and 4 factors in the denominator. The 4 in the
denominator cancel with 4 in the numerator leaving one in the numerator.
33. Power rule with negative exponents: Problem type 2
Simplify.
Write your answer using only positive exponents.
When a product is raised to a power, raise each factor to the power.
When a power is raised to a power, multiply the exponents.
Move the variables that have a negative exponent to the denominator.
34. Using the power, product, and quotient rules to simplify expressions with negative
exponents
Simplify.
Write your answer using only positive exponents.
This problem may be easier if you get rid of the negative exponents, first.
The 3 exponent means reciprocal.
Move the rest of the variables with negative exponents between the numerator and
denominator.
Raise everything in the first parentheses to the 3 power.
There are 3 factors in the denominator and 4 factors in the numerator. The 3 in
the denominator cancel with 3 in the numerator leaving one in the numerator.
Multiply the s by adding the exponents.
35. Ordering numbers with negative exponents
Order the expressions by choosing <, >, or =.
The negative exponent means reciprocal. These can be rewritten and then the order
can be determined.
<
>
<
36. Simplifying a polynomial expression
Simplify.
Change the subtraction to add the opposite. To find the opposite of a polynomial,
change the sign of each term in the polynomial.
The parentheses can be taken out and like terms combined.
37. Multiplying a monomial and a polynomial: Problem type 2
Multiply.
Simplify your answer as much as possible.
Distribute the multiply by
to everything in the parentheses.
38. Multiplying binomials: Problem type 3
Multiply.
Simplify your answer.
Use FOIL to multiply. Multiply the firsts, multiply the outsides, multiply the insides,
and multiply the lasts.
39. Degree and leading coefficient of a polynomial in one variable
What are the leading coefficient and degree of the polynomial?
The degree is the largest exponent and the leading coefficient is the coefficient of the
term with the largest degree.
Leading coefficient = 7
Degree = 5
40. Factoring a quadratic with leading coefficient 1
Factor.
To factor a quadratic with leading coefficient 1, find two numbers that multiply to give
the last term that also add to give the coefficient of the middle term.
Find two numbers that multiply to give 36 and that add to give 9. These numbers are
12 and 3. These are the two numbers in the factors.
The quadratic will factor as
( + 12)(  3)
41. Factoring a quadratic with leading coefficient greater than 1
Factor.
To factor a quadratic with leading coefficient greater than 1, find two factors that
multiply to give the first term and two numbers that multiply to give the last term.
This may not always give the correct answer, so you must check the answer.
Two factors that multiply to give
are 5 and . Two numbers that multiply to give
4 are 2 and 2.
This MAY factor as
(5 + 2)( + 2)
Check this by multiplying.
This checks. The factors are
(5 + 2)( + 2)
42. Greatest common factor of two monomials
Find the greatest common factor of the two expressions
and
Simplify your answer as much as possible.
Find the greatest common factor of the numbers then find the greatest common factor
of the variables.
The greatest common factor of 10 and 15 is 5.
The greatest common factor of the variables will be the smallest number of times it is
used in both.
43. Factoring out a monomial from a polynomial: Problem type 2
Factor the following expression.
The common monomial factor will be the greatest common factor of the two terms.
The greatest common factor is
.
Each term can be written as a multiple of the greatest common factor.
Use the distributive property to rewrite this.
44. Factoring a product of a quadratic trinomial and a monomial
Factor completely.
First, factor out a common monomial from the polynomial.
Factor the remaining polynomial. This is a quadratic with leading coefficient 1. Find
two numbers whose product is 25 and whose sum is 10. The two numbers are 5 and 5.
Use these to write the factors. Remember the 3 factor.
45. Multiplying rational expressions: Problem type 1
Multiply.
Simplify your answer as much as possible.
Cancel common factors before multiplying.
2 goes into both 4 and 2. 5 goes into both 5 and 25.
There is an
in both the numerator and denominator, these will cancel.
There is one in the numerator and 3 in the denominator; these will cancel leaving 2 in
the denominator.
There is another in the numerator and 2 in the denominator; these will cancel leaving
1 in the denominator.
Now multiply.
46. Adding rational expressions with common denominators
Subtract.
Simplify your answer as much as possible.
Change the subtraction to add the opposite. To find the opposite of a fraction, find the
opposite of the numerator. To do this, change the sign of each term in the numerator.
Now combine like terms in the numerator and keep the common denominator.
47. Dividing rational expressions: Problem type 2
Divide.
Simplify your answer as much as possible.
To divide, change to multiply by the reciprocal.
Now factor everything.
There is a common factor of 1 in the numerator and denominator that will cancel.
There is also a common factor of + 3 that will cancel.
Multiply these for the final answer.
48. Simplifying a ratio of polynomials: Problem type 1
Simplify.
Factor everything.
Now cancel common factors
49. Solving a rational equation that simplifies to a linear equation: Problem type 1
Solve for .
Simplify your answer as much as possible.
Whenever an equation has a fraction, multiply both sides by the denominator.
The s on the left will cancel.
Divide both sides by 9.
This will simplify.
50. Finding the roots of a quadratic equation with leading coefficient 1
Solve for .
First, factor the quadratic. Find two numbers whose product is 12 and whose sum is
4. These numbers are 6 and 4.
Set each factor equal to zero.
 6 = 0 and + 2 = 0
Solve these.
= 6 and = 2