Lesson 6: Negative and Fractional ExponentsExponents Review In Chapter 8, Lesson 3, we discussed the practical de-finition of exponentials: The expression x n means x multiplied by itself
Trang 1SAT Practice 5: Data Analysis
Questions 1 and 2 refer to the following information:
3. According to the graph above, Wacky Water Park experienced its largest increase in park at-tendance between which two consecutive months?
(A) March and April (B) April and May (C) May and June (D) June and July (E) July and August
Revenue per Unit
Cost per Unit Widgets
Gadgets Tinkers
$w
$x
$6
$4
$3
$y
COST/REVENUE ANALYSIS FOR THE TRINKET FACTORY
1. Which of the following best approximates the
slope m (in pounds per week) of the line that
best approximates these data?
(A) m > 1 (B) 0 < m < 1 (C) m= 1
(D) −1 < m < 0 (E) m < −1
2. If the line of best fit for the data presented
above passed through the points (32, 6.5) and
(39.5, 8.0), it can be estimated that a baby born
at 28 weeks would most nearly weigh how
many pounds?
(A) 5.3 (B) 5.5 (C) 5.7
60
45
30
15
0
MONTHLY TICKET SALES
AT WACKY WATER PARK
Mar Apr May Jun Jul Aug
4. The table above shows the per unit revenue and cost of three products at the Trinket Factory If profit equals revenue minus cost, how much profit do they make if they produce and sell two
of each item?
(A) 2w + 2x − 2y − 2 (B) 2y − 2x − 2w − 2 (C) w + x − y − 1 (D) x + 2w + y − 7 (E) 2x + 2y − 2w + 2
Trang 2Questions 5 and 6 refer to the following tables: 6. If the girls who bought turkey sandwiches have
$206 in total to spend on their lunches, what is the greatest number of turkey sandwiches with fries they could buy without exceeding their budget?
(A) 23 (B) 24 (C) 25 (D) 26 (E) 27
5. Based on the tables above, if every boy bought a
sandwich without fries and every girl bought a
sandwich with fries, how much more money did
the boys spend at the deli than the girls?
Turkey Ham Veggie Boys
Girls
Number of sandwiches ordered
at a local deli
Sandwich Sandwich
and Fries Turkey
Ham
Veggie
$ 4.50
$ 5.00
$ 3.75
$ 5.50
$ 6.00
$ 4.75 Cost of food
1
2
3
4
5
7
8
9
6
1
0
2
3
4
5
7
8
9
6
1
0
2
3
4
5
7
8
9
6
1 0
2 3 4 5
7 8 9 6
Trang 34 A The revenue generated from two widgets, two
gadgets, and two tinkers is $2w, $2x, and $2(6),
respectively The cost of producing two widgets,
two gadgets, and two tinkers is $8, $6, and $2y,
respectively Therefore, the total profit can be found by subtracting the cost from the revenue:
(2w + 2x + 12) − (8 + 6 + 2y) = 2w + 2x − 2y − 2.
5 86 The boys bought only sandwiches and spent
($4.50)(75) + ($5.00)(80) + $3.75(22) = $820.00 If the girls bought only sandwiches with fries, then they spent ($5.50)(40) + ($6.00)(35) + $4.75(64) =
$734.00 $820.00 − $734.00 = $86.00
6 D If x = sandwiches: x(4.5) + (40 − x)(5.5) = 206
Distribute: 4.5x + 220 − 5.5x = 206
Combine like terms: −1.0x = −14
There were (40 − x) = 40 − 14 = 26 meals with fries.
Concept Review 5
1 A best fit line is a straight line that “hugs” the data
most closely on a scatterplot
2 It can be created by connecting the “outermost”
points on the plot or by drawing a line that best
“hugs” the points and divides them in half Try to
ignore any outliers that don’t fit with the rest of
the data
3 For the SAT, you just need to be able to tell if a
slope is positive or negative, or perhaps greater or
less than 1 Positive slopes go up to the right, and
negative slopes go down to the right If the slope
is positive and the “rise” is greater than the “run,”
the slope is greater than 1; if the rise is less than
the run, the slope is less than 1
4 If you know what percent of the data are in a
sec-tor of the pie chart, multiply the percentage by
360° to obtain the degree measure of that sector
(e.g., a sector that represents 40% of the circle
would be (.40)(360°) = 144°)
5 The biggest percent change occurs between
Thursday and Fridays
Percent change = (100%)(10.5 − 5)/(5) = 110%
6 There were a total of 13 + 7 + 14 + 5 + 5 + 10.5 + 11.5 = 66,000 accidents; 35,000 of them occurred
on Friday, Saturday, or Sunday 35/66 = 53%
7 25% of the kids said blue was their favorite color 25% of 4,000 = 0.25 × 4,000 = 1,000 20% of the kids said yellow was their favorite color 20% of 4,000 = 0.20 × 4,000 = 800 1,000 − 800 = 200 kids
8 The color red represents 10% of the circle, which
is 10% of 360° °
9 There are currently (.25)(4,000) = 1,000 votes for blue In order for blue to be 50% of the circle, it would need (.50)(4,000) = 2,000 votes Therefore, 1,000 votes must change
10 Hamden High ordered 36 + 28 + 81 = 145 tickets New Haven Public ordered 64 + 23 + 64 = 151 Waterbury High ordered 53 + 31 + 51 = 135
11 81 Hamden High students bought 5-day passes at
$200.00/pass, spending (81)(200) = $16,200 64 students at New Haven Public bought 5-day passes at $200/pass, spending (64)(200) = $12,800 The difference is $16,200 − $12,800 = $3,400
10 360
( )=
Answer Key 5: Data Analysis
SAT Practice 5
1 B A line connecting (32, 6.5) and (39.5, 8.0) is a
good line of fit, and has a slope of 1.5/7.5 = 0.2,
which is between 0 and 1
2 C If the slope is about 0.2, you can use the slope
equation to solve:
Plug in values:
Simplify:
Multiply by 4: (6.5 − y1) = 0.8
Subtract 6.5: −y1= −5.7
Divide by −1: y1= 5.7
3 C From March to April: 30 − 15 = 15,000
From April to May: 30 − 30 = 0
From May to June: 60 − 30 = 30,000
From June to July: 65 − 60 = 5,000
From July to August: 69 − 65 = 4,000
6 5
1
−
( )y =
6 5
1
−
−
( y)=
y y
x x
0 2
−
−
Trang 4Lesson 6: Negative and Fractional Exponents
Exponents Review
In Chapter 8, Lesson 3, we discussed the practical
de-finition of exponentials:
The expression x n means x multiplied by itself
n times.
This is a useful definition when you need to evaluate
something like 43: you simply multiply 4 × 4 × 4 and
get 64 But what about expressions like 40or 4−3or
41/2? How do you multiply 4 by itself 0 times, or −3
times, or half of a time? It doesn’t make much sense
to think of it that way So to understand such
ex-pressions, you must expand your understanding of
exponents
Zero and Negative Exponents
Using what you have learned in Lesson 1 of this
chap-ter, what are the next three terms of this sequence?
81, 27, 9, 3, _, _, _
The rule seems to be “divide by 3,” so the next three
terms are 1, 1⁄3, and 1⁄9
Now, what are the next three terms of this sequence?
34, 33, 32, 31, _, _, _
Here, the rule seems to be “reduce the power by 1,” so
that the next three terms are 30, 3−1, and 3−2
Notice that the two sequences are exactly the same,
that is, 34= 81, 33= 27, 32= 9, and 31= 3 This means
that the pattern can help us to understand zero and
negative exponents: 30= 1, 3−1=1⁄3, and 3−2=1⁄9 Now,
here’s the million-dollar question:
Without a calculator, how do you write 3−7without a
negative exponent?
If you follow the pattern you should see that
and, in general:
x0 1 and
Notice that raising a positive number to a
neg-ative power does not produce a negneg-ative result.
For instance 3−2does not equal −9; it equals 1⁄9
Fractional Exponents
What if a number is raised to a fractional exponent?
For instance, what does 81/3mean? To understand expressions like this, you have to use the basic rules
of exponents from Chapter 8, Lesson 3 Specifically,
you need to remember that x n x m x m+n (81/3)3= 81/3× 81/3× 81/3 Using the rule above, 81/3× 81/3×
81/3= 81/3 + 1/3 + 1/3= 81= 8 In other words, when you raise
81/3to the 3rd power, the result is 8 This means that
81/3is the same as the cube root of 8, and, in general:
The expression x1/nmeans , or the nth root
of x For example, can be written as a1/2
Example:
What is the value of 163/4? The first step is to see that 163/4is the same as (161/4)3
(because (161/4)3= 161/4× 161/4× 161/4= 163/4) Using the definition above, 161/4is the 4th root of 16, which is 2 (because 24= 16) So (161/4)3= 23= 8
The expression x m/n means the nth root of x raised to the mth power For instance, 43/2
means the square root of 4 raised to the third power, or 23= 8
a
x
n
x x
n n
3
7 7
Trang 5Concept Review 6:
Negative and Fractional Exponents
Evaluate the following expressions without a calculator
Simplify the following expressions, eliminating any negative or fractional exponents
13 If x3/4= 27, what is the value of x?
14 If b−1/2= 4, what is the value of b?
15 If (2m)−6= 16, what is the value of 23m?
16 25
3 2
⎛
⎝⎜
⎞
⎠⎟
−
Trang 6SAT Practice 6:
Negative and Fractional Exponents
1 If 4n= 20, then what is the value of 4−n?
2 If 54× m = 52, then m=
(A) −52
(B) 5−2
(C)
(D) 51/2
(E)
3 If 2m× 2m× 2m× 2m = 2, then m =
1
2
1
5
4 For all values of n,
(A) 3 (B) (C) 3n
(D) 92
(E) 9n
5 If x is a positive number, then
(A) x3/4 (B) x–(1/4) (C) x3/4 (D) (E)
6 If x = a5= b3and x is positive, then ab= (A) x1/5
(B) x1/8
(C) x8/15
(D) x8
(E) x15
x5
x3
2 3
⎛
⎝⎜
⎞
⎠⎟
n
3 3 9
2
n
1
2
3
4
5
7
8
9
6
1
0
2
3
4
5
7
8
9
6
1
0
2
3
4
5
7
8
9
6
1 0
2 3 4 5
7 8 9 6
1
2
3
4
5
7
8
9
6
1
0
2
3
4
5
7
8
9
6
1
0
2
3
4
5
7
8
9
6
1 0
2 3 4 5
7 8 9 6
x3/2× x1/2
x –(1/2)
Trang 75 E
Simplify numerator (add exponents):
Simplify quotient (subtract exponents):
Rewrite as a root:
Solve for a (raise to the 1/5): x1/5= a Solve for b (raise to the 1/3): x1/3= b Multiply a and b: ab = x1/5× x1/3
Simplify (add exponents): ab = x1/5 + 1/3
(Remember the quick way to add fractions:
“zip-zap-zup” from Chapter 7, Lesson 3.)
x5
x4 2 − − ( 1 2 / )
x x
4 2
1 2
−
x x x
3 2 1 2 1 2
−
Concept Review 6
1 5−2= 1/(52) = 1/25
2
3 2−5= 1/(25) = 1/32
4
5
6 (16/25)−3/2= 1/(16/25)3/2= (25/16)3/2= ((25/16)1/2)3
7
8
9 4x−2= 4/x2
10 (4y)−2= 1/(4y)2= 1/(16y2)
11 9 3 2 9 1 2 3 27
m m m m m
( ) =⎛( )
4g1 2 4g 2 g
x1 3=3x
=⎛
⎝
⎠
⎟ =⎛⎝⎜ ⎞⎠⎟ =
25
16
5 4
125 64
43 2=( )41 23=( )43=23=8
25− 1 2 =1 25( )1 2 =1 25=1 5
91 2= 9=3
12
Raise both sides to the 4/3 power: (x3/4)4/3= 274/3
Simplify: x1= (271/3)4
Raise both sides to the
−2 power: (b−1/2)−2= 4−2
Simplify: b1= 1/(42)
Raise to the −1/2 power: 23m= 16−1/2
Simplify: 23m= 1/41 16
9
1
b b b b b b
b
( ) ( ) =( ) ×( ) = ×
=
−
Answer Key 6:
Negative and Fractional Exponents
SAT Practice 6
1 1/20 or 0.05 4−n= 1/4n= 1/20
Divide by 54: m= 52/54
Simplify (subtract exponents): m= 5−2
3 1/4 or 0.25 2m× 2m× 2m× 2m= 2
Simplify (add exponents): 24m= 2
Exponents must be equal: 4m= 1
4 A
Write as powers of 3:
Simplify denominator:
Divide numerator and
denominator by 32n: 31/1 = 3
Perhaps a simpler method is to simply pick n to be
0 (because n can be any number) This gives
(3 × 30)/90 = (3 × 1)/1 = 3 The only choice that
equals 3 when n= 0 is (A)
3
2
n
3
2
×
( )
n n
3 3 9
2
n
Trang 8WRITING A GREAT ESSAY
CHAPTER 12
437
✓ Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use
Trang 9Lesson 1: Map the SAT Essay Assignment
Consider carefully the issue discussed in the following
passage, and then write an essay that answers the
question posed in the assignment
Our leaders love to tell us that only victory will
do, as if they are imparting great wisdom They
seek to defeat the enemy, to achieve the goal
Yet many times a loss, particularly one that is
hard fought, is more valuable than victory We
cannot live a life full of only victories, nor
should we The quality of our lives depends as
much on how we manage our losses as on how
we achieve our victories
Quality is much more important than quantity, but it’s hard to get a great score with fewer than four paragraphs This is so because the readers
are looking for structure and development, which
require good use of paragraphs Think of your paragraphs as “stepping-stones” on a journey Only two or three stepping-stones don’t make for much of a journey Plan to write four well-defined paragraphs—five if you have enough time
The essay assignment asks you to formulate a
point of view regarding a particular aspect of
human values or behavior It does not require
you to recall any specific knowledge from your
studies, although you should try to connect
your thesis with your studies There is never a
“right” or “wrong” answer to the question; that
is, your actual position does not affect your
score More important (contrary to what a lot
of SAT-prep folks claim), the graders are not
looking for essays that fit a particular formula
You can use narration, exposition, persuasion,
or argument as long as it is focused on
devel-oping an interesting point of view that answers
the question
Assignment: Can a loss ever be more valuable
than a victory? Write an essay in which you answer
this question and discuss your point of view on this
issue Support your position logically with examples
from literature, the arts, history, politics, science
and technology, current events, or your experience
or observation
Know What They’re Looking For
Two English teachers who have been trained by the
Educational Testing Service (ETS) will read and
score your essay from 1 (poor) to 6 (outstanding)
They are trained to look for five things:
Interesting, relevant, and consistent point of view Do you take a thoughtful and interesting
position on the issue? Do you answer the ques-tion as it is presented? Do you maintain a con-sistent point of view?
Good reasoning Do you define any necessary
terms to make your reasoning clear? Do you explain the reasons for and implications of your thesis? Do you acknowledge and address possible objections to your thesis without sac-rificing its integrity?
Solid support Do you give relevant and specific
examples to support your thesis? Do you explain how these examples support your thesis?
Logical organization Does every paragraph
re-late clearly to your thesis? Do you provide logi-cal transitions between paragraphs? Do you have a clear introduction and conclusion? Does the conclusion provide thoughtful commentary rather than mere repetition of the thesis?
Effective use of language Do you use effective
and appropriate vocabulary? Do you vary sen-tence length and structure effectively? Do you avoid needless repetition? Do you use paral-lelism, metaphor, personification, or other rhetorical devices to good effect? Do you use strong verbs? Do you avoid needlessly abstract language? Do you avoid cliché?
The readers will not mark you down for minor spelling or grammar mistakes, and they won’t mark you up just for using big words Focus
on good reasoning If you can take an interesting
position, explore its implications, discuss relevant examples that support it, and maintain your focus, you will get a very good score
How Long Should It Be?
The Assignment
Your essay assignment will look something like this:
Trang 10Practice 1: Map the SAT Essay Assignment
SAT Essay Grading Review
1 What does it mean for an essay to have good substance?
2 What does it mean for an essay to have strong organization?
3 What does it mean for an essay to be clear?
4 What does it mean for an essay to have an effective and interesting style?
5 How long should your SAT essay be?
Check your answers with the answer key at the end of the chapter