Download Apply GCF and LCM to Monomials

Apply GCF and LCM to Monomials
Mississippi Standard: Apply the concepts of Greatest Common Factor (GCF)
and Least Common Multiple (LCM) to monomials with variables.
You can find the Greatest Common Factor (GCF) of two or more monomials by
finding the product of their common prime factors.
Find the GCF of Monomials
Find the GCF of 16xy2 and 30xy.
Find the prime factorization of each monomial.
16xy2 2 2 2 2 x y y
30xy 2 3 5 Circle the common factors.
xy
The GCF of 16xy2 and 30xy is 2 x y or 2xy.
You can find the Least Common Multiply (LCM) of two or more monomials by
multiplying the factors, using the common factors only once.
Find the LCM of Monomials
Find the LCM of 18xy2 and 10y.
Find the prime factorization of each monomial.
18xy2 2 3 3 10y
2
xyy
5
y
2335xyy
The LCM of
18xy2
Circle the common factors.
Multiply the factors, using the common factors only once.
and 10y is 2 3 3 5 x y y or 90xy2.
Exercises
Find the GCF of each set of monomials.
1. 12x, 40x2
2. 18m, 45mn
3. 14n, 42n2
4. 4st, 10s
5. 5ab, 6b2
6. 14b, 56b2
7. 36a3b, 56ab2
8. 30a3b2, 24a2b
9. 32mn2, 16n, 12n3
Find the LCM of each set of monomials.
10. 20c, 12c
11. 16xy, 3x
12. 36ab, 4b
13. 16a2,
14. 7x, 12x
15. 21mn, 28n2
17. 75n2, 25n4
18. 10x, 20x2, 40xy
14ab
16. 20st, 50s2t
GEOMETRY For Exercises 19 and 20, use the squares shown.
19. What is the GCF of the sides of the squares?
20. What is the LCM of the sides of the squares?
4xy
16x 2
Compare Data Sets
Mississippi Standard: Use a given mean, mode, median, and range to
summarize and compare data sets including investigation of the different
effects that change in data values have on these measures.
When given the mean, median, mode, and range of data sets, you can
often analyze and make comparisons of the data, without knowing the
data values.
Summarize and Compare Data Sets
BUILDINGS The mean, median, mode, and range of the ten tallest
buildings in New York and Texas are given in the table.
Ten Tallest Buildings (meters)
Mean
Median
Mode
Range
New York
Texas
289
281
248, 319
133
260
255
none
76
Which state has a greater average building height of its ten tallest
buildings?
The mean, or average, height of the ten tallest buildings in New York is
289 meters. The mean height of the ten tallest buildings in Texas is
260 meters. Compare the data.
289 260
So, New York has a greater average building height of its ten tallest
buildings than Texas.
Which state has the same building height for more than one of its ten
tallest buildings?
The mode represents data values that appear most often. In New York, a
height of 248 meters and 319 meters appear most often. In Texas, there
are no building heights that appear more than once or most often. So,
New York has the same building height for more than one of its ten
tallest buildings.
The tallest building in Texas is 305 meters. How tall is the tenth tallest
building?
The range of the ten tallest buildings in Texas is 76 meters, so subtract
76 from 305.
305 76 229
The tenth tallest building in Texas is 229 meters tall.
When a value of a data set changes, you can often determine how the
mean, median, mode, and range will be affected without recalculating all
the measures.
Investigate a Change in Data Value
SCHOOL The table at the right shows the math test scores of
Fifth Period
Period Test
Test Scores
Scores (%)
(%)
Fifth
Mr. Gomez’s fifth period class. The mean is 81%, the median
is 82%, the mode is 85%, and the range is 30%. If after handing
back the tests, the student who received a 65% did extra credit
to change his test score to 75%, how will this affect the mean,
median, mode, and range?
Since the test score is increasing, the mean will also increase.
90
82
85
70
75
91
95
70
72
72
85
85
70
78
90
65
95
88
82
75
78
80
82
85
85
The median is the middle number. Since 75% is still in the lower
part of the test scores, it will not affect the median.
The mode is the number that appears most often. A new score of 75% will
result in three test scores of 75%. However, there will still be five test scores
of 85%, so the mode will stay the same.
The range is the difference between the greatest score and the least score.
The greatest score will still be 95%. However, the least score will now be
70%. So, the range will be 95% 70%, or 25%, instead of 30%.
Exercises
BASEBALL For Exercises 1–4, use the information in the
Number of Runs Batted In by
Season Leader, 1997–2006
table on the number of runs batted in by the season leader
for the National League and American League.
1. Which league had a greater average number of runs
National
League
American
League
142.9
144
147
32
147.7
146
145
28
Mean
Median
Mode
Range
batted in by its season leaders from 1997–2006?
2. From 1997–2006, the greatest number of runs batted
in by the season leader for the National League was
160 runs. What was the fewest number of runs batted
in by the season leader?
Source: The World Almanac
3. From 1997–2006, the fewest number of runs batted in by the season leader
for the American League was 137 runs. What was the greatest number of
runs batted in by the season leader?
4. Write a statement comparing the middle number of runs batted in by the
National League and American League season leaders from 1997–2006.
ONLINE For Exercises 5 and 6, use the list below that shows the number of
hours Ms. Wright’s students spent online last week. The mean of the data is
6, the median is 7, the mode is 7, and the range is 12.
7
4
7
9
3
1
5
8
10
0
9
4
0
11
3
1
9
0
7
11
12
7
6
8
4
10
5. If the student who spent 12 hours online thought about it more and
changes his or her number of hours to 9 hours, how will this affect the
mean, median, mode, and range?
6. If Ms. Wright recorded a student who said 1 hour as 11 hours on the list,
how will this affect the mean, median, mode, and range when she corrects
the data value to 1 hour and recalculates the measures?
Multiply and Divide with Scientific Notation
Mississippi Standard: Multiply and divide numbers written in scientific notation.
You can use scientific notation to simplify computations with very large and/or
very small numbers.
To multiply numbers in scientific notation, regroup to multiply the factors and
multiply the powers of ten. Then simplify. To multiply the powers of ten, use the
Product of Powers .
Product of Powers
Words
To multiply powers with the same base, add their exponents.
Symbols
Arithmetic
32
35
32 5
Algebra
or
37
xa
xb x a b
Multiplication with Scientific Notation
Evaluate the expression (1.3 102)(2.5 101).
(1.3 102)(2.5 101) (1.3 2.5)(102 101)
(3.25)(102 101)
3.25 Commutative and Associative Properties
Multiply 1.3 by 2.5.
102 1
Product of Powers
3.25 103
Add the exponents.
3.25 1,000
103 1,000
3,250
Move the decimal point 3 places.
Evaluate the expression (4.2 103)(1.6 104).
(4.2 103)(1.6 104) (4.2 1.6)(103 104)
(6.72)(103 104)
6.72 Commutative and Associative Properties
Multiply 4.2 by 1.6.
103 4
Product of Powers
6.72 107
Add the exponents.
6.72 10,000,000
107 10,000,000
67,200,000
Move the decimal point 7 places.
To divide numbers in scientific notation, regroup to divide the factors and
divide the powers of ten. Then simplify. To divide the powers of ten, use the
Quotient of Powers .
Quotient of Powers
Words
To divide powers with the same base, subtract their exponents.
Symbols
Arithmetic
48
43
48 3
603 Prerequisite Skills
or
Algebra
45
xa
xa b, x
xb
0
Division with Scientific Notation
9.45 106
2.1 10
Evaluate the expression 3 .
Associative Property
10
4.5 3
10 Divide 9.45 by 2.1.
4.5 106 3
Quotient of Powers
4.5 103
Subtract the exponents.
4.5 1,000
103 1,000
4,500
Move the decimal point 3 places.
9.45 106
9.45 106
3
2.1 10
2.1 103
6
Exercises
Multiply or divide. Express using exponents.
1. 51 54
25
2
4. 3
2. 65 64
3. 102 103
76
7
109
10
5. 5
6. 6
Evaluate each expression. Express the result in scientific notation and
standard form.
7. (2.6 105)(1.9 102)
8. (5.3 104)(0.9 103)
9. (3.7 102)(1.2 102)
10. (3.3 103)(2.1 102)
11. (8.5 103)(1.1 101)
12. (3.9 102)(2.3 106)
13. (6.45 105)(1.2 103)
14. (4.18 104)(0.9 105)
15. 3
8.37 108
2.7 10
16. 2
17. 5
9.72 109
1.8 10
18. 3
8.32 107
1.3 10
20. 6
4.2 108
1.68 10
22. 8
19. 5
21. 2
8.04 105
6.7 10
4.64 104
2.9 10
6.3 1010
1.5 10
9.0 1011
2.4 10
23. BASEBALL The table shows the 2007 salaries of
six Major League Baseball players. About how
many times greater is Alex Rodriguez’s salary
than Juan Castro’s salary?
24. ASTRONOMY The Sun burns about
4.4 106 tons of hydrogen per second. How
much hydrogen does the Sun burn in one year?
(Hint: one year 3.16 107 seconds)
25. OCEANS The area of the Pacific Ocean is
6.0 107 square miles. The area of the Atlantic
Ocean is 2.96 107 square miles. About how many
times greater is the area of the Pacific Ocean than
the Atlantic Ocean?
2007 Major League Baseball Salaries
Player
Team
Juan Castro
Coco Crisp
Nomar Garciaparra
Chipper Jones
Kazuo Matsui
Alex Rodriguez
Cincinnati Reds
Boston Red Sox
Los Angeles Dodgers
Atlanta Braves
Colorado Rockies
New York Yankees
Source: USA Today
Salary
(dollars)
9.25 105
3.83 106
8.52 106
1.23 107
1.5 106
2.27 107
The Density Property
Mississippi Standard: Develop a logical argument to demonstrate the
‘denseness’ of rational numbers.
Examine the number line below. Find another integer that lies between the
integers 2 and 3.
0
-2 -1
1
2
3
4
The integers 1, 0, 1, and 2 all lie between 2 and 3 on the number line.
On the number line above, find a number that lies between 1 and 2. Their
1
2
average, 1, is one number that lies between 1 and 2.
Find a Number Between Two Given Numbers
1
3
1
2
Find a number that lies between and on the number line below.
0
1
6
1
3
2
3
1
2
5
6
1
One number would be their average.
1 1
1
1 2
3
2 3
2
2 6
6
1 5
1
1
Rewrite and with a common denominator.
3
2
Add the numerators.
2 6
5
12
Multiply.
1
3
5
12
1
2
The rational number, , lies between and .
Find a number that lies between 7 and 6.5.
One number would be their average.
1
1
[7 (6.5)] (13.5)
2
2
6.75
Add 7 and 6.5.
Multiply.
The rational number, 6.75, lies between 7 and 6.5.
The process above of finding another number between any two given numbers
can be continued indefinitely. This suggests the density property .
Density Property for Rational Numbers
Words
Between every pair of distinct rational numbers, there are infinitely
many rational numbers.
You can use the density property to solve real-world problems.
Apply the Density Property
BAKING Genevieve’s grandmother gave her a family recipe for apple pie.
Her grandmother does not use an exact amount of sugar, but told
1
4
1
2
Genevieve to use somewhere between 1 and 1 cups of sugar. If
Genevieve wants to use an exact amount of sugar that is somewhere
1
4
1
2
between 1 cups and 1 cups, how much sugar can she use?
One possible amount is their average.
1 3
2
2 4
1 11
2 4
1 1
1
1 1
2
1 1 1 1
2 4
2
2 4
4
11
8
1
2
Rename as .
2
4
Add the whole numbers and add the fractions.
3
Rewrite 2 as an improper fraction.
4
3
8
or 1
Simplify.
3
8
So, Genevieve can use 1 cups of sugar.
Exercises
Identify a number that lies between points A and B on each number line.
1.
0
1
A
B
2
3
3.
0
0.5
1
1.5
2.
4
5
A
B
2
2.5
A
B
-4 -3 -2 -1
4.
1
10
2
10
A
B
3
10
4
10
0
1
5
10
6
10
Identify a number that lies between each pair of numbers.
3
4
6. 10 and 9
5. 6 and 7
1
2
8. 2 and 1
11. 4 and 3
2
3
7. and 1
3
4
1
3
10. 5 and 4
9. 4 and 4
12. 8.25 and 8.75
13. 15.5 and 16
14. SCHOOL For reading class, Dylan is recording the number of hours he reads
1
2
each week. This week, Dylan needs to read between 1 and 2 hours. What
1
2
is a possible time between 1 and 2 hours that Dylan can read?
15. CROSS COUNTRY For cross-country practice, the coach told the runners
1
2
3
4
they needed to run between 5 and 5 miles. Give a possible distance between
1
2
3
4
5 and 5 miles that a runner can run.
16. Demonstrate the density property for rational numbers with several
examples of your own.
Algebraic Properties
Mississippi Standard: Apply algebraic properties in problem-solving.
Review the properties in the table below. These properties can be applied when
problem-solving.
Properties
Commutative Property
sum or product.
The order in which numbers are added or multiplied does not change the
6776
3883
Examples
abba
abba
The way in which numbers are grouped when added or multiplied does not
Associative Property
change the sum or product.
(2 7) 4 2 (7 4)
(3 4) 5 3 (4 5)
Examples
(a b) c a (b c)
(a b) c a (b c)
To multiply a sum by a number, multiply each addend by the number
Distributive Property
outside the parentheses.
2(7 4) 2 7 2 4
(5 6)3 5 3 6 3
Examples
Additive Identity
a(b c) ab ac
(b c)a ba ca
The sum of any number and 0 is the number.
707
Examples
a0a
The product of any number and 0 is 0.
Zero Property of Multiplication
900
Examples
Multiplicative Identity
a00
The product of any number and 1 is the number.
313
Examples
a1a
Identify Properties
Name the property shown by each statement.
(7 3x) 2x 7 (3x 2x)
0 5a 0
Associative Property
Zero Property of Multiplication
Use Properties to Simplify Expressions
Simplify each expression. Justify each step.
4 (x 13)
4 (x 13) 4 (13 x)
Commutative Property
(4 13) x
Associative Property
17 x
Add 4 and 13.
6(x 7)
6(x 7) 6(x) 6(7)
6x 42
Distributive Property
Multiply.
Apply Properties to Problem Solving
MUSEUMS Three friends are going to the science museum. The cost of
admission is $x each. It will cost an additional $4 to view a movie on the
3-D screen. Write and simplify an expression that represents the total cost
for the three friends.
The cost of admission plus the movie can be represented by (x + 4).
Multiply this cost by the number of friends, 3(x + 4).
3(x 4) 3(x) 3(4)
3x 12
Distributive Property
Multiply.
So, the total cost for the three friends is $3x $12.
MUSEUMS Refer to Example 5. A fourth friend will meet the group of
friends at the museum but will not go to the movie. Write and simplify
an expression that represents the total cost for the four friends.
The cost for the fourth friend is $x. Add this to $3x $12.
3x 12 x 3x x 12
4x 12
Commutative Property
Add.
So, the total cost for the four friends is $4x $12.
Exercises
Name the property shown by each statement.
1. 3n m m 3n
2. 0 18d 18d
3. (7y 8) 10y 7y (8 10y)
4. 20xy 1 20xy
5. 3(6a 7b) 3 6a 3 7b
6. 82 0 0
Simplify each expression. Justify each step.
7. 1 (6 x)
8. 5(6a)
9. 11 (6 n)
10. 5(x 8)
11. 15(4w)
12. 9(x 2)
13. 9 2y 11 5y
14. 4(x 7) 2x
15. 11n 7(2 3n)
16. ANIMALS A zebra can run up to 40 miles per hour. An elephant can run up to
x miles per hour. Write and simplify an expression to find how many more miles a
zebra will run in six hours than an elephant.
17. CELL PHONES Seven friends have similar cell phone plans. The price of each
plan is $x. Three of the seven friends pay an extra $4 per month for unlimited
text messaging. Write and simplify an expression that represents the total cost of
the seven plans.
Make Predictions from Circle Graphs
and Histograms
Mississippi Standard: Use proportions, estimates, and percentages to
construct, interpret, and make predictions about a population based on
histograms or circle graph representations of data from a sample.
You can make predictions about a given set of data displayed in a circle graph
or histogram. Use percentages to make predictions about data displayed in a
circle graph.
Favorite Type of Television Show
Predict from a Circle Graph
4%
5% Other
Drama
The circle graph shows the results of a survey of
the students in the 8th grade at Oakwood Junior
High. If there are 560 students at Oakwood Junior
High, how many would you predict to choose
reality as their favorite type of television show?
7%
Fiction
40%
Reality
16%
Cartoon
The section of the graph representing students who
chose reality is 40% of the circle. So find 40% of 560.
28%
Comedy
To find 40% of 560, you can use either method.
METHOD 1 Write the percent as a decimal.
40% of 560 40% 560
Write a multiplication expression.
0.40 560
Write 40% as a decimal.
224
Multiply.
METHOD 2 Write the percent as a fraction.
40% of 560 40% 560
40
560
100
1
224
Write a multiplication expression.
560
1
Write 40% as a fraction. Write 560 as .
Multiply.
So, about 224 students at Oakwood Junior High would choose
reality as their favorite type of television show.
Predict from a Histogram
The bar at 43.0–44.9 seconds is much higher than
the others and represents the most winning times.
So, the winning speed of the runner in the next
summer Olympics will most likely be in the
43.0–44.9 second range.
Summer Olympic Men’s 400-Meter Run
Winning Times, 1896–2004
Number of Winners
The histogram shows the winning times of the
men’s 400-meter run in the summer Olympics
from 1896 to 2004. Predict the range of speeds
that a runner finishing in first place is most
likely to be in the next summer Olympics?
Explain your reasoning.
12
10
8
6
4
2
0
43.0– 45.0– 47.0– 49.0– 51.0– 53.0–
44.9 46.9 48.9 50.9 52.9 54.9
Time (seconds)
Source: The World Almanac
Most Popular
Luxury
Car Colors
5%
Exercises
CARS For Exercises 1–3, use the circle graph that shows the most
popular luxury car colors.
Other
9%
Blue
1. If a car dealership sold 50 luxury cars in March, predict how
many were white.
26%
Silver/
Gray
9%
Red
2. If a car dealership sold 250 luxury cars in January through
11%
Light
Brown
June, predict how many were black.
3. If a parking garage has 85 luxury cars parked on a given day,
28%
White
12%
Black
predict how many are silver/gray.
Favorite Summertime Activity
VACATION For Exercises 4 and 5, use the circle graph that
5%
6% Other
Read
shows the results of a survey of the favorite summertime
activities of 7th grade students at Parson Junior High.
4. If there are 275 students at Parson Junior High, predict
how many would choose visiting an amusement park as
their favorite summertime activity.
32%
Amusement
Park
13%
Beach
5. If there are 150 students at Parson Junior High, predict
how many would choose swimming or going to camp as
their favorite summertime activity.
20%
Camp
HISTORY For Exercises 6 and 7, use the histogram
U.S. Presidents Age at Inauguration
that shows the age of U.S. presidents at their
inauguration.
10
8
6
4
2
9
4
9
4
–6
65
–6
60
–5
55
–5
50
–4
40
9
0
4
president will most likely be in at their
inauguration.
12
–4
7. Predict the 10-year age range that the next U.S.
14
45
president will most likely be in at their
inauguration.
16
Number of Presidents
6. Predict the 5-year age range that the next U.S.
24%
Swim
Age at Inaguration
8. Predict the range that students in Mrs. Jeng’s
second period math class will most likely score.
9. Predict the range that students in Mrs. Jeng’s third
period math class will least likely score.
Mrs. Jeng’s First Period Test Scores
14
Number of Students
SCHOOL For Exercises 8 and 9, use the histogram that
shows the test scores of Mrs. Jeng’s first period math
class. Mrs. Jeng teaches three math classes of the
same level in the morning.
12
10
8
6
4
2
0
51–60 61–70 71–80 81–90 91–100
Score