Cracking the SAT with 5 Practice Tests, 2014 Edition (2013)
Part III. How to Crack the Math Section
Chapter 11. Fun with Fundamentals
Although we’ll show you which mathematical concepts are most important to know for the SAT, this book relies on your knowledge of basic math concepts. If you’re a little rusty, this chapter is for you. Read on for a quick review of the math fundamentals you’ll need to know before you continue.
THE BUILDING BLOCKS
As you go through this book you might discover that you’re having trouble with stuff you thought you already knew—like fractions, or square roots. If this happens, it’s probably a good idea to review the fundamentals. That’s where this chapter comes in. Our drills and examples will refresh your memory if you’ve gotten rusty, but if you have serious difficulty with the following chapters, even after reviewing the material in this chapter, then you should consider getting extra help. For this purpose, we recommend our own Math Workout for the SAT, which is designed to give you a thorough review of all the fundamental math concepts that you’ll need to know on the SAT. Always keep in mind that the math tested on the SAT is different from the math taught in school. If you want to raise your score, don’t waste time studying math that ETS never tests.
Let’s talk first about what you should expect to see on the test.
THE MATH BREAKDOWN
Three of the nine scored sections on the SAT are Math sections. Two of the scored Math sections will last 25 minutes each; the third will last 20 minutes.
No Need to Know
Here are a few things
that you won’t need to
know to answer SAT
math questions: calculus,
trigonometry, and the
quadratic formula.
Essentially, the SAT tests
the math you learned in
junior high and your first
two years of high school.
The math questions on your SAT will be drawn from the following four categories:
1. Arithmetic
2. Basic Algebra I and II
3. Geometry
4. Basic probability/statistics
That’s it! Although there are some smaller, miscellaneous topics that will occasionally appear on the SAT (logic, visual perception, permutations/combinations), they are very, very rare: don’t expect to see more than 1 or 2 of those questions on the test.
The math questions on the SAT are almost entirely arithmetic, basic algebra, or geometry.
The math questions on your SAT will appear in two different formats:
1. Regular multiple-choice questions
2. Grid-Ins
The Grid-Ins will only appear in one of the scored 25-minute sections. All other math questions will be normal multiple-choice questions.
THE INSTRUCTIONS
Each of the three scored Math sections on your SAT will begin with the same set of instructions. We’ve reprinted these instructions, just as they appear on the SAT, in the Math sections of the practice tests in this book. These instructions include a few formulas and other information that you may need to know in order to answer some of the questions. You should learn these formulas ahead of time so you don’t have to waste valuable time referring to them during the test.
Still, if you do suddenly blank out on one of the formulas while taking the test, you can always refresh your memory by glancing back at the instructions. Be sure to familiarize yourself with them thoroughly ahead of time, so you’ll know which formulas are there.
BASIC PRINCIPLES OF SAT NUMBERS
Before moving on, you should be certain that you are familiar with some basic terms and concepts that you’ll need to know for the Math sections of the SAT. This material isn’t at all difficult, but you must know it cold. If you don’t, you’ll waste valuable time on the test and lose points that you easily could have earned.
Positive and Negative
There are three rules regarding the multiplication of positive and negative numbers.
1. positive × positive = positive
2. negative × negative = positive
3. positive × negative = negative
Integers
Integers are the numbers that most of us are accustomed to thinking of simply as “numbers.” Integers are numbers that have no fractional or decimal part. They can be either positive or negative. The positive integers are
1, 2, 3, 4, 5, 6, 7, and so on
The negative integers are
–1, –2, –3, –4, –5, –6, –7, and so on
Zero (0) is also an integer, but it is neither positive nor negative.
Note that positive integers get bigger as they move away from 0, while negative integers get smaller. In other words, 2 is bigger than 1, but –2 is smaller than –1.
Negative Land
Think of integers as steps
on a staircase leading
up from the cellar (the
negatives), through a
doorway (zero), and above
the ground (the positives).
Five steps down (–5) is
farther below ground
than four steps down (–4)
because you’re one step
farther away from the
cellar door (0). Integers are
like stairs, because when
climbing stairs, you can’t
use a fraction of a step.
This number line should give you a clear idea of how negative numbers work. On the number line below, as you go to the right, the numbers get larger. As you go to the left, the numbers get smaller (–4 is smaller than –3, which is smaller than –2, and so on).
You should also remember the types of numbers that are not integers. Here are some examples:
Basically, integers are numbers that have no fractions or decimals. So if you see a number with a fraction or non-zero decimal, it’s not an integer.
Odd and Even
Even numbers are integers that can be divided by 2 leaving no remainder. Here are some examples of even numbers:
–4, –2, 0, 2, 4, 6, 8, 10, and so on
You can always tell at a glance whether a number is even: It is even if its final digit is even (divisible by 2). Thus 999,999,999,994 is an even number because 4, the final digit, is an even number.
Odd numbers are integers that have a remainder when divided by 2. Here are some examples of odd numbers:
–5, –3, –1, 1, 3, 5, 7, 9, and so on
You can always tell at a glance whether a number is odd: It is odd if its final digit is odd. Thus, 444,444,444,449 is an odd number because 9, the final digit, is an odd number.
Pick a Number, Any Number…
If you aren’t sure about one of these rules, it’s safer to try an example than guess. Say you need to know what kind of number you get when you add an odd number and an even number to solve a problem but you can’t remember the rule. Just try an example like 2 + 5 = 7 and you’ll know that even + odd = odd.
Several rules always hold true with odd and even numbers:
even + even = even
odd + odd = even
even + odd = odd
even × even = even
odd × odd = odd
even × odd = even
Distinct Numbers
You might see problems on the SAT that mention “distinct numbers.” Don’t let this throw you. All ETS means by distinct numbers is different numbers. For example, the list of numbers 2, 3, 4, and 5 is a set of distinct numbers, whereas 2, 2, 3, and 4 would not be a list of distinct numbers because 2 appears twice. Easy concept, tricky wording.
Digits
There are 10 digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
All integers are made up of digits. In the integer 3,476, the digits are 3, 4, 7, and 6. Digits are to numbers as letters are to words.
The integer 645 is called a “three-digit number” for obvious reasons. Each of its digits has a different name depending on its place in the number:
5 is called the units digit.
4 is called the tens digit.
6 is called the hundreds digit.
By the way, you may be more used to referring to the digit as the “ones digit.” It’s the exact same idea (the last number before the decimal point), but for some reason ETS calls the ones digit the units digit.
But what if we didn’t use an integer? If we had a number such as 645.32, which contains decimal places, we would know that it’s a little bit bigger than 645. Each decimal place tells us how much bigger.
3 is called the tenths digit.
2 is called the hundredths digit.
Thus the value of any number depends on which digits are in which places. The number 645.32 could be rewritten as follows:
Factors
The factors of an integer are all of the integers that divide into it evenly. For example, the factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.
Which Is Which?
The largest factor of a
number is that number.
The smallest multiple of a
number is that number.
Multiples
A multiple of a number is any product of an integer and the given number. For example, 10, 20, 50, 180, and 370 are all multiples of 10. Make sure you know the difference between a factor and a multiple.
Remainders
If an integer cannot be divided evenly by another number, the integer left over at the end of the division is called the remainder. Decimals cannot be remainders.
Leftovers
Be careful when figuring
remainders on your
calculator. On your
calculator, 25 divided
by 3 is 8.3333333, but
.3333333 is not the
remainder. The remainder
is 1, or .33333333 of 3.
So do the long division
instead!
The best way to figure out a remainder is to actually do the long division. For example, if you want to find the remainder when 25 is divided by 3, set up and start solving a long division problem. Here’s what it would look like:
The 1 that is left over after you subtract the 24 is the remainder.
Consecutive Integers
Consecutive integers are integers listed in increasing order of size without any integers missing in between. For example, –1, 0, 1, 2, 3, 4, and 5 are consecutive integers; 2, 4, 5, 7, and 8 are not. Nor are –1, –2, –3, and –4 consecutive integers, because they are decreasing in size.
Prime Numbers
A prime number is a positive integer that is divisible only by itself and by 1.
Here are a few important facts about prime numbers:
· 0 and 1 are not prime numbers.
· 2 is the smallest prime number.
· 2 is the only even prime number.
· Not all odd numbers are prime: 1, 9, 15, 21, and many others are not prime.
The Prime Directive
The SAT frequently asks questions about the prime numbers smaller than 30, so make sure you have these memorized: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
Also, notice that 1 is not prime? There will often be a trap answer choice on SAT questions that assumes that you think that 1 is a prime number. So remember: 1 is NOT prime.
Standard Symbols
The following standard symbols are used frequently on the SAT:
SYMBOL |
MEANING |
= |
is equal to |
≠ |
is not equal to |
< |
is less than |
> |
is greater than |
≤ |
is less than or equal to |
≥ |
is greater than or equal to |
THERE ARE ONLY SIX OPERATIONS
There are only six arithmetic operations that you will ever need to perform on the SAT:
1. Addition (3 + 3)
2. Subtraction (3 – 3)
3. Multiplication (3 × 3 or 3 • 3)
4. Division (3 ÷ 3)
5. Raising to a power (3^{3})
6. Finding a square root ()
If you’re like most students, you probably haven’t paid much serious attention to these topics since junior high school. You’ll need to learn about them again if you want to do well on the SAT. By the time you take the test, using them should be automatic. All the arithmetic concepts are fairly basic, but you’ll have to know them cold. You’ll also have to know when and how to use your calculator, which will be quite helpful.
In this chapter, we’ll deal with each of these six topics.
What Do You Get?
You should know the following arithmetic terms:
· The result of addition is a sum or total.
· The result of subtraction is a difference.
· The result of multiplication is a product.
· The result of division is a quotient.
· In the expression 5^{2}, the 2 is called an exponent.
The Six Operations Must Be Performed in the Proper Order
Very often, solving an equation on the SAT will require you to perform several different operations, one after another. These operations must be performed in the proper order. In general, the problems are written in such a way that you won’t have trouble deciding what comes first. In cases in which you are uncertain, you need to remember only the following sentence:
Please Excuse My Dear Aunt Sally;
she limps from left to right.
That’s PEMDAS, for short. It stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. First, do any calculations inside the parentheses; then you take care of the exponents; then you perform all multiplication and division, from left to right, followed by addition and subtraction, from left to right.
Do It Yourself
Some calculators
automatically take order
of operations into account,
and some don’t. Either
way, you can very easily
go wrong if you are in
the habit of punching in
long lines of arithmetic
operations. The safe,
smart way is to clear the
calculator after every
individual operation,
performing PEMDAS
yourself.
The following drill will help you learn the order in which to perform the six operations. First, set up the equations on paper. Then, use your calculator for the arithmetic. Make sure you perform the operations in the correct order.
Whichever Comes First
For addition and subtraction,
solve from left to
right. The same is true of
multiplication and division.
And remember: If you
don’t solve in order from
left to right, you
could end up with the
wrong answer!
Example:
24 ÷ 4 × 6 = 24 ÷ 24 = 1
wrong
24 ÷ 4 × 6 = 6 × 6 = 36
right
DRILL 1
Solve each of the following problems by performing the indicated operations in the proper order. Answers can be found on this page.
1. 107 + (109 – 107) = _____________
2. (7 × 5) + 3 = _____________
3. 6 – 3 (6 – 3) = _____________
4. 2 × [7 – (6 ÷ 3)] = _____________
5. 10 – (9 – 8 – 6) = _____________
Parentheses Can Help You Solve Equations
Using parentheses to regroup information in SAT arithmetic problems can be very helpful. In order to do this, you need to understand a basic law that you have probably forgotten since the days when you last took arithmetic—the distributive law. You don’t need to remember the name of the law, but you do need to know how it works.
The Distributive Law
If you’re multiplying the sum of two numbers by a third number, you can multiply each number in your sum individually. This comes in handy when you have to multiply the sum of two variables.
If a problem gives you information in “factored form”—a (b + c)—then you should distribute the first variable before you do anything else. If you are given information that has already been distributed—(ab + ac)—then you should factor out the common term, putting the information back in factored form. Very often on the SAT, simply doing this will enable you to spot ETS’s answer.
Here are some examples:
Distributive: 6(53) + 6(47) = 6(53 + 47) = 6(100) = 600
Multiplication first: 6(53) + 6(47) = 318 + 282 = 600
You get the same answer each way, so why get involved with ugly arithmetic? If you use the distributive law for this problem, you don’t even need to use your calculator.
The drill on the following page illustrates the distributive law.
DRILL 2
Rewrite each problem by either distributing or factoring (Hint: for questions 1, 2, 4, and 5, try factoring) then solve. Questions 3, 4, and 5 have no numbers in them; therefore, they can’t be solved with a calculator. Answers can be found on this page.
1. (6 × 57) + (6 × 13) = ____________
2. 51(48) + 51(50) + 51(52) = ________
3. a(b + c – d) = _____________
4. xy – xz = _____________
5. abc + xyc = _____________
FRACTIONS
A Fraction Is Just Another Way of Expressing Division
The expression is exactly the same thing as x ÷ y. The expression means nothing more than 1 ÷ 2. In the fraction , x is known as the numerator (hereafter referred to as “the top”) and y is known as the denominator (hereafter referred to as “the bottom”).
Fractions and Your Calculator
You can also use your calculator to solve fraction problems. When you do, ALWAYS put each of your fractions in a set of parentheses. This will ensure that your calculator knows that they are fractions. Otherwise, the order of operations will get confused. On a scientific calculator, you can write the fraction in two different ways:
You will have a fraction key, which looks similar to “a^{b}/_{c}.” If you wanted to write , you’d type “5 a^{b}/_{c} 6.” You can also use the division key, because a fraction bar is the same as “divided by.” Be aware that your answer will be a decimal for this second way, so we recommend the first.
On a graphing calculator, you’ll use the division bar to create fractions. Keep in mind that, whatever calculator you are using, you can always turn your fractions into decimals before you perform calculations with them. Just be aware that the answer won’t always be exact.
Adding and Subtracting Fractions with the Same Bottom
To add two or more fractions that all have the same bottom, simply add up the tops and put the sum over the common bottom. Consider this example:
Subtraction works exactly the same way:
Adding and Subtracting Fractions with Different Bottoms
In school you were taught to add and subtract fractions with different bottoms, or denominators, by finding a common bottom. To do this, you have to multiply each fraction by a number that makes all the bottoms the same. Most students find this process annoying.
Calculating Fractions
Let’s say you wanted to find = using your calculator. For a scientific calculator, you’d type in “(1 a^{b}/_{c} 3) + (1 a^{b}/_{c} 2) =” The answer will come up looking like something similar to 5|6, which means 5/6. On a graphing calculator, you’d type in (1/3) + (1/2) [ENTER]. This gives you the repeating decimal .833333. Now hit the [MATH] button and hit the [>FRAC] button and press [ENTER]. The calculator will now show “5/6.” The shortcut to turn a decimal into a fraction on a TI-80 series graphic calculator is [MATH][ENTER][ENTER]. Remember those parentheses for all fraction calculations!
Fortunately, we have an approach to adding and subtracting fractions with different bottoms that simplifies the entire process. Use the example below as a model. Just multiply in the direction of each arrow, and then either add or subtract across the top. Lastly, multiply across the bottom.
We call this procedure the Bowtie because the arrows make it look like a bowtie. Use the Bowtie to add or subtract any pair of fractions without thinking about the common bottom, just by following the steps above.
Multiplying All Fractions
Multiplying fractions is easy. Just multiply across the top, then multiply across the bottom.
Here’s an example:
When you multiply fractions, all you are really doing is performing one multiplication problem on top of another.
You should never multiply two fractions before looking to see if you can reduce either or both. If you reduce first, your final answer will be in the form for which ETS is looking. Here’s another way to express this rule: Simplify before you multiply.
Dividing All Fractions
To divide one fraction by another, flip over (or invert) the second fraction and multiply. Doing this is extremely easy, as long as you remember how it works.
Just Flip It
When dividing (don’t ask
why) just flip the last one
and multiply.
Here’s an example:
Be careful not to cancel or reduce until after you flip the second fraction. You can even do the same thing with fractions whose tops and/or bottoms are fractions. These problems look quite frightening but they’re actually easy if you keep your cool.
Here’s an example:
Reducing Fractions
When you add or multiply fractions, you will very often end up with a big fraction that is hard to work with. You can almost always reduce such a fraction into one that is easier to handle.
Start Small
It is not easy to see that
26 and 286 have a common
factor of 13, but it’s
pretty clear that they’re
both divisible by 2.
So start from there.
To reduce a fraction, divide both the top and the bottom by the largest number that is a factor of both. For example, to reduce , divide both the top and the bottom by 12, which is the largest number that is a factor of both. Dividing 12 by 12 yields 1; dividing 60 by 12 yields 5. The reduced fraction is .
If you can’t immediately find the largest number that is a factor of both, find any number that is a factor of both and divide both the top and bottom by that. Your calculations will take a little longer, but you’ll end up in the same place. In the previous example, even if you don’t see that 12 is a factor of both 12 and 60, you can no doubt see that 6 is a factor of both. Dividing top and bottom by 6 yields . Now divide by 2. Doing so yields . Once again, you have arrived at ETS’s answer.
Fast Reduction
If you are calculator savvy,
reducing fractions will be a
breeze. To reduce fractions
in your scientific calculator,
just type in the fraction and
hit the equals key. If you
are using a graphing calculator,
type in the fraction,
find the [>FRAC] function,
and hit enter.
Converting Mixed Numbers to Fractions
A mixed number is a number such as 2. It is the sum of an integer and a fraction. When you see mixed numbers on the SAT, you should usually convert them to ordinary fractions.
Here’s a quick and easy way to convert mixed numbers.
· Multiply the integer by the bottom of the fraction.
· Add this product to the top of the fraction.
· Place this sum over the bottom of the fraction.
For example, let’s convert 2 to a fraction. Multiply 2 (the integer part of the mixed number) by 4 (the bottom of the fraction). That gives you 8. Add that to the 3 (the top of the fraction) to get 11. Place 11 over 4 to get .
The mixed number 2 is exactly the same as the fraction . We converted the mixed number to a fraction because fractions are easier to work with than mixed numbers.
DRILL 3
Try converting the following mixed numbers. Answers can be found on this page.
1.
2.
3.
4.
5.
Just Don’t Mix
For some reason, ETS
thinks it’s okay to give
you mixed numbers as
answer choices. On grid-ins,
however, if you use a
mixed number, ETS won’t
give you credit. You can
see why. In your grid-in
box 3¼ will be gridded in
as 3 1/4, which looks like
; not the answer you
thought you were using,
was it?
Fractions Behave in Peculiar Ways
Joe Bloggs has trouble with fractions because they don’t always behave the way he thinks they ought to. For example, because 4 is obviously greater than 2, Joe Bloggs sometimes forgets that is less than . He becomes especially confused when the top is some number other than 1. For example, is less than .
Joe also has a hard time understanding that when you multiply one fraction by another, you will get a fraction that is smaller than either of the first two. Study this example:
A Final Word About Fractions and Calculators
Throughout this section we’ve given you some hints about your calculator and fractions. While you should understand how to work with fractions the old-fashioned way, your calculator can be a tremendous help if you know how to use it properly. Make sure that you practice with your calculator so that working with fractions on it becomes second nature before the test.
DRILL 4
Work these problems with the techniques you’ve read about in this chapter so far. Then check your answers by solving them with your calculator. If you have any problems, go back and review the information just outlined. Answers can be found on this page.
1. Reduce . ____________________
2. Convert 6 to a fraction. ____________________
3. = ____________________
4. = ____________________
5. = ____________________
6. = ____________________
7. = ____________________
DECIMALS
A Decimal Is Just Another Way of Expressing a Fraction
Fractions can be expressed as decimals. To find a fraction’s decimal equivalent, simply divide the top by the bottom. (You can do this easily with your calculator.)
Adding, Subtracting, Multiplying, and Dividing Decimals
Manipulating decimals is easy with a calculator. Simply punch in the numbers—being especially careful to get the decimal point in the right place every single time—and read the result from the display. A calculator makes these operations easy. In fact, working with decimals is one area on the SAT where your calculator will prevent you from making careless errors. You won’t have to line up decimal points or remember what happens when you divide. The calculator will keep track of everything for you, as long as you punch in the correct numbers to begin with. Just be sure to practice carefully before you go to the test center.
DRILL 5
Answers can be found on this page.
1. 0.43 × 0.87 = ________________
2. = ________________
3. 3.72 ÷ 0.02 = ________________
4. 0.71 – 3.6 = ________________
Comparing Decimals
Some SAT problems will ask you to determine whether one decimal is larger or smaller than another. Many students have trouble doing this. It isn’t difficult, though, and you will do fine as long as you remember to line up the decimal points and fill in missing zeros.
Place Value
Compare decimals place
by place, going from left
to right.
Here’s an example:
Problem: Which is larger, 0.0099 or 0.01?
Solution: Simply place one decimal over the other with the decimal points lined up, like this:
0.0099
0.01
To make the solution seem clearer, you can add two zeros to the right of 0.01. (You can always add zeros to the right of a decimal without changing its value.) Now you have this:
0.0099
0.0100
Which decimal is larger? Clearly, 0.0100 is, just as 100 is larger than 99. (Remember that 0.0099 = , while 0.0100 = . Now the answer seems obvious, doesn’t it?)
Analysis
Joe Bloggs has a terrible time on this problem. Because 99 is obviously larger than 1, he tends to think that 0.0099 must be larger than 0.01. But it isn’t. Don’t get sloppy on problems like this! ETS loves to trip up Joe Bloggs with decimals. In fact, any time you encounter a problem involving the comparison of decimals, you should stop and ask yourself whether you are about to make a Joe Bloggs mistake.
EXPONENTS AND SQUARE ROOTS
Exponents Are a Kind of Shorthand
Many numbers are the product of the same factor multiplied over and over again. For example, 32 = 2 × 2 × 2 × 2 × 2. Another way to write this would be 32 = 2^{5}, or “thirty-two equals two to the fifth power.” The little number, or exponent, denotes the number of times that 2 is to be used as a factor. In the same way, 10^{3} = 10 × 10 × 10, or 1,000, or “ten to the third power,” or “ten cubed.” In this example, the 10 is called the base and the 3 is called the exponent. (You won’t need to know these terms on the SAT, but you will need to know them to follow our explanations.)
Exponents and Your Calculator
Raising a number to a power is shown in two different ways on your calculator, depending on the type of calculator you have. A scientific calculator will use the y^{x} button. You’ll have to type in your base number first, then hit the y^{x} key, then type the exponent. So 4^{3} will be typed in as “4 y^{x} 3 =” and you’ll get 64. If you have a calculator from the TI-80 series, your button will be a ^ sign. You’ll enter the same problem as “4^3 [ENTER].” Think of these two keys as the “to the” button, because you say “4 to the 3rd power.”
Multiplying Numbers with Exponents
When you multiply two numbers with the same base, you simply add the exponents. For example, 2^{3} × 2^{5} = 2^{3+5} = 2^{8}.
Warning
The rules for multiplying
and dividing exponents do
not apply to addition
or subtraction:
2^{2} + 2^{3} = 12
(2 × 2) + (2 × 2 × 2) = 12
It does not equal 2^{5} or 32.
Dividing Numbers with Exponents
When you divide two numbers with the same base, you simply subtract the exponents. For example, = 2^{5−3} = 2^{3}.
Raising a Power to a Power
When you raise a power to a power, you multiply the exponents. For example, (2^{3})^{4} = 2^{3 × 4} = 2^{12}
Warning
Parentheses are very
important with exponents,
because you must
remember to distribute
powers to everything
within them. For example,
(3x)^{2} = 9x^{2}, not 3x^{2}.
Similarly, , not
. But the distribution rule
applies only when you
multiply or divide. (x + y)^{2} =
x^{2} + 2xy + y^{2}, not x^{2} + y^{2}.
MADSPM
To remember the exponent rules, all you need to do is remember the acronym MADSPM. Here’s what it stands for:
· Multiply → Add
· Divide → Subtract
· Power → Multiply
Whenever you see an exponent problem, you should think MADSPM. The three MADSPM rules are the only rules that apply to exponents.
Here’s a typical ETS exponent problem:
14. For the equations = a^{10} and, if (a^{y})^{3} = a > 1, what is the value of x ?
(A) 5
(B) 10
(C) 15
(D) 20
(E) 25
Here’s How to Crack It
This problem looks pretty intimidating with all those variables. In fact, you might be about to cry “POOD” and go on to the next problem. That might not be a bad idea but before you skip the question, pull out those MADSPM rules.
For the first equation, you can use the Divide-Subtract rule: = a^{x − y} = a^{10}. In other words, the first equation tells you that x − y = 10.
For the second equation, you can use the Power-Multiply rule: (a^{y})^{3} = a^{3y} = a^{x}. So, that means that 3y = x.
Now, it’s time to substitute: x − y = 3y − y = 10. So, 2y = 10 and y = 5. Be careful, though! Don’t choose answer A. That’s the value of y, but the question wants to know the value of x. Since x = 3y, x = 3(5) = 15, which is answer C.
Don’t forget that you could also do this question by plugging in the answer choices, which will be discussed near the end of Chapter 12 (Algebra: Cracking the System). Of course, you still need to know the MADSPM rules to do the question that way.
Calculator Exponents
You can compute simple exponents on your calculator. Make sure you have a scientific calculator with a y^{x} key. To find 2^{10}, for example, simply use your y^{x} key, punching 2 in for the y value and 10 in for the x value.
The Peculiar Behavior of Exponents
Raising a number to a power can have quite peculiar and unexpected results, depending on what sort of number you start out with. Here are some examples.
· If you square or cube a number greater than 1, it becomes larger. For example, 2^{3} = 8.
· If you square or cube a positive fraction smaller than one, it becomes smaller.
For example, .
· A negative number raised to an even power becomes positive.
For example, (–2)^{2} = 4.
· A negative number raised to an odd power remains negative.
For example, (–2)^{3} = –8.
See the Trap
ETS is hoping that you
won’t know these strange
facts about exponents,
so the test writers will
throw them in as trap
answers. Knowing the
peculiar behavior of
exponents will help you
avoid these tricky pitfalls
in a question.
You should also have a feel for relative sizes of exponential numbers without calculating them. For example, 2^{10} is much larger than 10^{2}. (2^{10} = 1,024; 10^{2} = 100.) To take another example, 2^{5} is twice as large as 2^{4}, even though 5 seems only a bit larger than 4.
Square Roots
The radical sign (√) indicates the square root of a number. For example, = 5. Note that square roots cannot be negative. If ETS wants you to think about a negative solution they’ll say x^{2}= 25 because then x = 5 or x = –5.
The Only Rules You Need to Know
Here are the only rules regarding square roots that you need to know for the SAT:
1. . For example, = 6.
2. . For example, .
3. = positive root only. For example, = 4.
Note that rule 1 works in reverse: . This is really a kind of factoring. You are using rule 1 to factor a large, clumsy radical into numbers that are easier to work with. And remember that radicals are just fractional exponents, so the same rules of distribution apply.
Careless Errors
Don’t make careless mistakes. Remember that the square root of a number between 0 and 1 is larger than the original number. For example, , and .
Roots and Your Calculator
The other important key is the root key. On a scientific calculator it is often the same button as y^{x}, but you’ll have to hit shift first. The symbol is . So “the 4th root of 81” would be “81 4 = .” Sometimes the calculator will have y^{x} or as x^{y} or . They mean the same thing. Just know which number you’re supposed to type in first.
The root key in the TI-80 graphing calculator series varies, but the most common symbol is the square root sign, which you can get to by pressing “[SHIFT] x^{2}.” In case you want to find the 3^{rd}, 4^{th}, or other root of a number, there is a button in the [MATH] directory for or . In the case of the , you have to type in the root you want, then hit [MATH] and , and finally hit your base number. For example, if you wanted to find the 4th root of 81, you’d type “4 [MATH],” then select , then type 81 and press [ENTER]. If you look at it on the screen, it will appear as “4 81,” which is similar to how you’d write it. You can also use the ^ symbol if you remember that a root is the same as the bottom part of a fractional exponent. We’ll go over this in more detail in the fractions section.
Negative and Fractional Exponents
So far we’ve dealt with only positive integers for exponents, but they can be negative integers as well as fractions. The same concepts and rules apply, but the numbers just look a little weirder. Keep these concepts in mind:
· Negative exponents are a fancy way of writing reciprocals:
· Fractional exponents are a fancy way of taking roots and powers:
Here’s an example:
18. If x > 0, which of the following is equivalent to ?
I. x +
II.
III.
(A) None
(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II and III
Here’s How to Crack It
This problem really tests your knowledge of exponents. First, convert into an exponent since all the roman numerals contain expressions with exponents. (Plus, exponents are easier to work with because they have those nice MADSPM rules.) So, using the definition of a fractional exponent, = . You want the items in the roman numerals to equal .
Now, it’s time to start working with the roman numerals. For roman numeral I, ETS is trying to be tricky. (There’s a surprise.) There’s no exponent rule for adding exponent expressions with like bases. So, . (If you were uncertain, you could try a number for x. If x = 4, then = 8 but .) Cross off any answer with roman numeral I. So, B, C, and E are all gone.
Now, all you really need to do is try either II or III. If either works, the answer is D, since you are down to either answer choice A, None, and answer choice D, II and III. For roman numeral II, use the power-multiply rule: . So, since roman numeral II works, D is the correct answer.
If you simply must know what’s going on with roman numeral III, use the multiply-add rule: . But, remember that you had already chosen an answer. Using good POE on a roman numeral question often means that you don’t need to check all the roman numerals.
Summary
· There are only six arithmetic operations tested on the SAT: addition, subtraction, multiplication, division, exponents, and square roots.
· These operations must be performed in the proper order (PEMDAS), beginning with operations inside parentheses.
· Apply the distributive law whenever possible. Very often, this is enough to find ETS’s answer.
· A fraction is just another way of expressing division.
· You must know how to add, subtract, multiply, and divide fractions. Don’t forget that you can also use your calculator.
· If any problems involving large or confusing fractions appear, try to reduce the fractions first. Before you multiply two fractions, for example, see if it’s possible to reduce either or both of the fractions.
· If you know how to work out fractions on your calculator, use it to help you with questions that involve fractions. If you intend to use your calculator for fractions, make sure you practice. You should also know how to work with fractions the old-fashioned way.
· A decimal is just another way of expressing a fraction.
· Use a calculator to add, subtract, multiply, and divide decimals.
· Exponents are a kind of shorthand for expressing numbers that are the product of the same factor multiplied over and over again.
· To multiply two exponential expressions with the same base, add the exponents.
· To divide two exponential expressions with the same base, subtract the exponents.
· To raise one exponential expression to another power, multiply the exponents.
· To remember the exponent rules, think MADSPM.
· When you raise a positive number greater than 1 to a power greater than 1, the result is larger. When you raise a positive fraction less than 1 to an exponent greater than 1, the result is smaller. A negative number raised to an even power becomes positive. A negative number raised to an odd power remains negative.
· When you’re asked for the square root of any number , you’re being asked for the positive root only.
· Here are the only rules regarding square roots that you need to know for the SAT: