Determine whether each expression is a monomial. Write yes or no. Explain your reasoning.
1.15
SOLUTION:
15 is a monomial. It is a constant and all constants are monomials.
2.2− 3a
SOLUTION:
2− 3a is not a monomial. There is subtraction and more than one term.
3.
SOLUTION:
is not a monomial. There is a variable in the denominator.
4.−15g2
SOLUTION:
−15g2 is a monomial. It is the product of a number and variables.
5.
SOLUTION:
is a monomial. It is the product of a number and a variable.
6.7b + 9
SOLUTION:
No; there is addition and more than one term.
Simplify each expression.
7.k(k3)
SOLUTION:
8.m4(m2)
SOLUTION:
9.2q2(9q4)
SOLUTION:
10.(5u4v)(7u4v3)
SOLUTION:
11.[(32)2]2
SOLUTION:
12.(xy4)6
SOLUTION:
13.(4a4b9c)2
SOLUTION:
14.(−2f 2g3h2)3
SOLUTION:
15.(−3p5t6)4
SOLUTION:
16.GEOMETRY The formula for the surface area of a cube is SA = 6s2, where SA is the surface area and s is the
length of any side.
a. Express the surface area of the cube as a monomial.
b. What is the surface area of the cube if a = 3 and b = 4?
SOLUTION:
a.
b.
So the surface area is 69,984 square units.
Simplify each expression.
17.(5x2y)2(2xy3z)3(4xyz)
SOLUTION:
18.(−3d2f3g)2[(−3d2f)3]2
SOLUTION:
19.(−2g3h)(−3gj4)2(−ghj)2
SOLUTION:
20.(−7ab4c)3[(2a2c)2]3
SOLUTION:
Determine whether each expression is a monomial. Write yes or no. Explain your reasoning.
21.122
SOLUTION:
122 is a monomial. A constant is a monomial.
22.3a4
SOLUTION:
3a4 isamonomial.Itistheproductofanumberandvariables.
23.2c + 2
SOLUTION:
2c + 2 is not a monomial. It involves addition and has more than one term.
24.
SOLUTION:
is not a monomial. The expression has a variable in the denominator.
25.
SOLUTION:
is a monomial. It can be written as the product of a number and variable.
26.6m + 3n
SOLUTION:
6m + 3n is not a monomial. In the expression, there is addition and more than one term.
Simplify each expression.
27.(q2)(2q4)
SOLUTION:
28.(−2u2)(6u6)
SOLUTION:
29.(9w2x8)(w6x4)
SOLUTION:
30.(y6z9)(6y4z2)
SOLUTION:
31.(b8c6d5)(7b6c2d)
SOLUTION:
32.(14fg2h2)(−3f4g2h2)
SOLUTION:
33.(j5k7)4
SOLUTION:
34.(n3p)4
SOLUTION:
35.[(22)2]2
SOLUTION:
36.[(32)2]4
SOLUTION:
37.[(4r2t)3]2
SOLUTION:
38.[(−2xy2)3]2
SOLUTION:
GEOMETRYExpresstheareaofeachtriangleasamonomial.
39.
SOLUTION:
40.
SOLUTION:
Simplify each expression.
41.(2a3)4(a3)3
SOLUTION:
42.(c3)2(−3c5)2
SOLUTION:
43.(2gh4)3[(−2g4h)3]2
SOLUTION:
44.(5k2m)3[(4km4)2]2
SOLUTION:
45.(p5r2)4(−7p3r4)2(6pr3)
SOLUTION:
46.(5x2y)2(2xy3z)3(4xyz)
SOLUTION:
47.(5a2b3c4)(6a3b4c2)
SOLUTION:
48.(10xy5z3)(3x4y6z3)
SOLUTION:
49.(0.5x3)2
SOLUTION:
50.(0.4h5)3
SOLUTION:
51.
SOLUTION:
52.
SOLUTION:
53.(8y3)(−3x2y2)
SOLUTION:
54. (49m)(17p)
SOLUTION:
55.(−3r3w4)3(2rw)2(−3r2)3(4rw2)3(2r2w3)4
SOLUTION:
56.(3ab2c)2(−2a2b4)2(a4c2)3(a2b4c5)2(2a3b2c4)3
SOLUTION:
57.FINANCIALLITERACY Cleavon has money in an account that earns 3% simple interest. The formula for
computing simple interest is I = Prt, where I is the interest earned, P represents the principal that he put into the
account, r is the interest rate (in decimal form), and t represents time in years.
a. Cleavon makes a deposit of $2c and leaves it for 2 years. Write a monomial that represents the interest earned.
b. If c represents a birthday gift of $250, how much will Cleavon have in this account after 2 years?
SOLUTION:
a.
b.
Cleavon will make $30 interest, so he will have 250 + 30 = $280 in his account.
CCSSTOOLSExpress the volume of each solid as a monomial.
58.
SOLUTION:
59.
SOLUTION:
60.
SOLUTION:
61.PACKAGING For a commercial art class, Aiko must design a new container for individually wrapped pieces of
candy. The shape that she chose is a cylinder. The formula for the volume of a cylinder is V = πr2h.
a. The radius that Aiko would like to use is 2p3, and the height is 4p3. Write a monomial that represents the volume
of her container.
b. Make a table of five possible measures for the radius and height of a cylinder having the same volume.
c. What is the volume of Aiko’s container if the height is doubled?
SOLUTION:
a.
b. The product of the square of the radius’coefficient and the height’s coefficient must be 16. The exponents of the
radius and the height must have a sum of 9. Sample answer:
c. If the height is doubled, h = 2(4p3) = 8p3
So, the volume of Aiko’s container is 32πp9 cubic units.
62.ENERGY Albert Einstein’s formula E = mc2 shows that if mass is accelerated enough, it can be
converted into usable energy. Energy E is measured in joules, mass m in kilograms, and the speed of light is about
300 million meters per second.
a. Complete the calculations to convert 3 kilograms of gasoline completely into energy.
b. What is the energy if the amount of gasoline is doubled?
SOLUTION:
a.
So, 3 kilograms of gasoline converts to 270,000,000,000,000,000 joules of energy.
b. If the amount of gasoline is doubled to 6 kilograms, then the energy is also doubled.
63.MULTIPLEREPRESENTATIONS In this problem, you will explore exponents.
a.TABULAR Copy and use a calculator to complete the table.
b.ANALYTICAL What do you think the values of 50 and 5−1 are? Verify your conjecture using a calculator.
c.ANALYTICAL Complete: For any nonzero number a and any integer n, a−n = ______.
d.VERBAL Describe the value of a nonzero number raised to the zero power.
SOLUTION:
a.
b. If 30 = 1, then 50 may also equal 1. If 3-1 equals one-third, then 5-1 may equal one-fifth.
c. From the table, it seems that a number raised to a negative power is the same the same as its reciprocal raised to
the same power, only positive. So,
d. Any nonzero number raised to the zero power is 1.
64.CCSSPERSEVERANCE For any nonzero real numbers a and b and any integers m and t, simplify the expression
and describe each step.
SOLUTION:
Move the negative sign to the numerator. Since raising the fraction to the 2t power means multiplying both the
numerator and the denominator by themselves 2t times, we can rewrite the expression as the power of a power for
boththenumeratorandthedenominator.
To find the power of the power, multiply the exponents.
Regroupthenumeratortoisolatethenegativebase.
Simplify the numerator and denominator.
Use the power of a power rule one last time to simplify.
65.REASONING Copy the table below.
a. For each equation, write the related expression and record the power of x.
b. Graph each equation using a graphing calculator.
c. Classify each graph as linear or nonlinear.
d. Explain how to determine whether an equation, or its related expression, is linear or nonlinear without graphing.
SOLUTION:
a.
b.
c.
d. If the power of x is 1, the equation or its related expression is linear.
66.OPENENDED Write three different expressions that can be simplified to x6.
SOLUTION:
Sample answer: x4 ⋅ x2; x5 ⋅ x;Forthese,theexponentsmusthaveasumof6.
(x3)2; For this type, the product of 3 and 2 is 6.
67.WRITINGINMATH Write two formulas that have monomial expressions in them. Explain how each is used in a
real-world situation.
SOLUTION:
Sample answer: The area of a circle or A = πr2, where r is the radius, can be used to find the area of any circle. The
area of a rectangle or A = w ⋅ ℓ, where w is the width and ℓ is the length, can be used to find the area of any
rectangle.
68.Which of the following is not a monomial?
A −6xy
B
C
D 5gh4
SOLUTION:
The only example that has a negative exponent is C. This means that the term has a variable in the denominator.
Therefore it is not a monomial and choice C is the correct answer.
69.GEOMETRY The accompanying diagram shows the transformation of ΔXYZ to ΔX′Y′Z′ .
This transformation is an example of a
F dilation
G line reflection
H rotation
J translation
SOLUTION:
Because the original triangle got smaller in the transformation, it is a dilation. None of the other types of
transformations involve changing the size of a shape (they only affect location). Therefore, choice F is the correct
answer.
70.CARS In 2002, the average price of a new domestic car was $19,126. In 2008, the average price was $28,715.
Based on a linear model, what is the predicted average price for 2014?
A $45,495
B$38,304
C $35,906
D $26,317
SOLUTION:
Let x be the year and let y be the average price of a new domestic car. Since all the points are part of a linear
model, the slope between any two points will be the same.Find the slope of the line containing the data points (2002,
19,126) and (2008, 28,715) .
Use this slope and the points (2002, 19,126) and (2014, y2) to determine the average cost of a new domestic car
in 2014.
Thepredictedaveragepriceofanewdomesticcarfor2014isabout$38,304.Therefore,thecorrectchoiceisB.
71.SHORTRESPONSE If a line has a positive slope and a negative y–intercept, what happens to the x–intercept if
the slope and the y–intercept are doubled?
SOLUTION:
The y–intercept of the graphed equation is –4 and the slope is 2, so the equation would be . From the
graph, we see that the x–intercept is 2. Now, double the slope and intercept, and the new equation is .
Substitute y = 0, to find the x–intercept.
Notice that the x–intercept does not change.
Solve each system of inequalities by graphing.
72.y < 4x
2x + 3y≥ −21
SOLUTION:
Writeequation2inslope,interceptform.
Graph each inequality. The graph of y < 4x is dashed and is not included in the graph of the solution. The graph of 2x
+3y≥ −21 is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in
the intersection of the graphs of y < 4x and 2x+3y≥ −21. This region is darkly shaded in the graph below.
73.y ≥2
2y + 2x≤4
SOLUTION:
Write equation 2 in slope-intercept form.
Graph each inequality. The graph of y≥2issolidandisnotincludedinthegraphofthesolution.Thegraphof2y +
2x≤4isalsosolidandisincludedinthegraphofthesolution.Thesolutionofthesystemisthesetoforderedpairsin
the intersection of the graphs of y≥2and2y + 2x≤4.Thisregionisdarklyshadedinthegraphbelow.
74.y > −2x − 1
2y≤3x + 2
SOLUTION:
Rewrite equation 2 in slope-intercept form.
Graph each inequality. The graph of y > −2x − 1 is dashed and is not included in the graph of the solution. The graph
of 2y≤3x + 2 is solid and is included in the graph of the solution. The solution of the system is the set of ordered
pairs in the intersection of the graphs of y > −2x − 1 and 2y≤3x + 2. This region is darkly shaded in the graph
below.
75.3x + 2y < 10
2x + 12y < –6
SOLUTION:
Write each equation in slope-intercept form.
Equation1:
Equation2:
Graph each inequality. The graph of 3x + 2y < 10 is dashed and is not included in the graph of the solution. The
graph of 2x + 12y < –6 is also dashed and is not included in the graph of the solution. The solution of the system is
the set of ordered pairs in the intersection of the graphs of 3x + 2y < 10 and 2x + 12y < –6. This region is darkly
shaded in the graph below.
76.SPORTS In the 2006 Winter Olympic Games, the total number of gold and silver medals won by the U.S. was 18.
The total points scored for gold and silver medals was 45. Write and solve a system of equations to find how many
gold and silver medals were won by the U.S.
SOLUTION:
Each gold medal is worth 3 points and each silver medal is worth 2 points. So, if the U.S. won g gold medals and s
silver medals, then they won 3g + 2s points.
The total number of gold and silver medals, g + s, is 18. So, we have 2 equations:
Solve the 2nd equation for g.
Substitute for g in the other equation.
g = 18 – 9 = 9
g = 9; s = 9
77.DRIVING Tires should be kept within 2 pounds per square inch (psi) of the manufacturer’s recommended tire
pressure. If the recommendation for a tire is 30 psi, what is the range of acceptable pressures?
SOLUTION:
30 + 2 = 32
30 - 2 = 28
The pressure will range between 28 and 32 psi, inclusive.
78.BABYSITTING Alexis charges $10 plus $4 per hour to babysit. Alexis needs at least $40 more to buy a television
for which she is saving. Write an inequality for this situation. Will she be able to get her television if she babysits for
5 hours?
SOLUTION:
Let h represent the number of hours that Alexis has to babysit.
If she babysits 5 hours, substitute h = 5 into the inequality.
This inequality is not true, so she will not be able to afford her television if she babysits for 5 hours.
Find each quotient.
79.−64÷(−8)
SOLUTION:
The quotient of two negative integers is positive.
−64÷(−8) = 8
80.−78÷1.3
SOLUTION:
The quotient of a negative number and a positive number is negative.
−78÷1.3=–60
81.42.3÷(−6)
SOLUTION:
The quotient of a positive number and a negative number is negative.
42.3÷(−6) = –7.05
82.−23.94÷10.5
SOLUTION:
The quotient of a negative number and a positive number is negative.
−23.94÷10.5=–2.28
83.−32.5÷(−2.5)
SOLUTION:
The quotient of two negative numbers is positive.
−32.5÷(−2.5) = 13
84.−98.44÷4.6
SOLUTION:
The quotient of a negative number and a positive number is negative.
−98.44÷4.6=–21.4
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7-1 Multiplication Properties of Exponents
