Questions involve
-
Rules of exponents
- Factoring and simplifying algebraic expressions
- Understanding concepts of relations and
functions
- Solving first and second degree equations
and inequalities
- Solving simultaneous equations
- Setting up equations to solve word problems
- Applying basic algebra skills to solve
problems
Exponents
and Radicals
Many numbers can
be expressed as the product of a number multiplied by itself a number of
times. For example, 16 can be expressed as . Another way to write this is 24.
The 4 is called
the exponent and the 2 is called the base. The expression 24 is read "2
to the fourth power".
Here is a list
of some definitions you need to know in order to follow this discussion of
algebra. However, you will not need to know the definitions on the GRE.
1. Variable: A letter that represents an unknown number.
2. Term: A product of any combination of variables and numbers. For example,
2x, 3xy2, 10y3, etc. are all terms.
3. Coefficient: A number or variable in a term. For example, in the term
2xy2, 2 is
a coefficient of xy2, x is a coefficient of 2y2, y2 is a coefficient
of 2x, etc.
4. Expression: Any number of terms combined by addition or subtraction
signs.
For example, 3xyz + 6x2y – 10z is an algebraic expression.
5. Equation: Two expressions or terms set equal to each other. Do not
get an
expression and an equation confused. An equation has an equal sign in it; and an
expression does not.
An easy way to
remember the definitions of term, equation, and expression is to think of an
equation as the algebraic equivalent of a sentence and an expression as the algebraic equivalent
of a phrase. Then a term is just one word in the sentence or phrase.
When multiplying
terms with the same base, just add the exponents. For example, x5x 3 =
x8. However, x5y3(xy)8 because x5 and y3
do not have the same base. Do not make the mistake
of adding the exponents when multiplying numbers with different bases. Another common error
is adding the exponents when adding two terms with like bases. This rule does not apply to
addition. For example, 24 + 22 26.
When dividing
terms with the same base, just subtract the exponents. For example, 35 33
=
32. Again, this rule does not apply to division of terms with different bases or to subtraction
of
terms with like bases. For example, x5 y3 (x / y)2and 24 –
2222.
When adding
or subtracting two or more terms with exponents the terms must have like
bases and like exponents. If the bases and exponents are just alike, simply add the numerical
coefficients. Remember that if there is no numerical coefficient, it is understood to be 1. For
example, x2 + 5x2 – 2x2 = 4x2.
When raising
an exponent to another exponent, simply multiply the exponents. For
example, (x5)2 = x10. If there is a term with several coefficients
in the parentheses, you must
distribute the exponent to every coefficient. For example, (2xy2)3 = 8x3y6
and (3/2)2 = 9/4. This
rule does not apply if there is an addition or subtraction sign inside the parentheses. For
example, (2x + 3y)2 4x2 + 9y2 , this is a very common mistake, so be
careful not to make it!
The following is a
list of characteristics of exponents that you should commit to memory for
the GRE:
1. Any number raised to 0 is always 1. For example, 50 = 1, x0 =
1, etc.
2. Any number without an exponent is understood to have an exponent of 1. For
example,
x = x1, 2 = 21, etc.
3. Raising a number greater than 1 to a power greater than 1 results in a bigger
number.
For example, 22 = 4.
4. Raising a fraction between 0 and 1 to a power greater than 1 results in a
smaller fraction.
For example, (1/2)2 = 1/4. Recall from our discussion on fractions that we said multiplying
a fraction by another fraction results in a smaller fraction. Raising a number to a power is
equivalent to multiplying that number by itself.
5. A negative number raised to an even power results in a positive number. For
example, (-
3)2 = 9.
6. A negative number raised to an odd power results in a negative number. For
example, (-
2)3 = -8.
Although negative
exponents, such as 2-3, are a very important concept in your algebra class,
you will NOT see them on the GRE, so don’t worry about them.
You should have
a feel for the relative size of exponents. Remember that they are just
shorthand notation for multiplication. So, 25 is twice as large as 24. And 210
is more than 10
times as large as 102 (Why?).
The radical
sign indicates the square root of the number under the radical. Similarly, the sign
indicates the cube root of the number under the radical. If x2 = 16, then x = + 4
and + 4 are
called the positive and negative square roots of the number 16. However, = +4. In other
words, the radical sign only refers to the positive root of the number under the radical. Hence, if
the GRE asks for , the answer is +5, not –5.
Unlike for an algebra
class, there are only two radical rules you need to know for the
GRE:
Algebra
You are required
to know very little real algebra for the GRE. Algebra methods learned in your
algebra classes will often mislead you and will usually take up way too much time. For instance,
never try to set up an algebraic and work through to an answer. The GRE only cares which
space you blacken on the answer sheet, not how you work an algebra problem. In this section,
we will discuss the small subset of algebra rules that you actually need to know to do well on
the GRE.
There are a few
more definitions that you will need to know in order to follow this section.
However, again, you do not need to know these for the GRE.
1. Binomial: An algebraic
expression containing 2 terms.
2. Trinomial: An algebraic expression containing 3 terms.
3. Polynomial: A binomial, trinomial, or any other algebraic expression
containing two or
more terms.
The following are
some helpful hints to remember for the GRE. See Review #1 if you need to
review factoring and "unfactoring" (i.e., the distributive laws).
1. When you encounter
a problem containing an expression that can be factored,you should
always factor that expression. For example, if you see an expression such as 4x + 4y, you
should immediately factor it into 4(x + y).
2. Similarly, whenever
you see an expression that has been factored, you should immediately
"unfactor" it, i.e. "multiply it out". For example, if a problem contains the expression
4(x + y),
multiply it out, to get 4x + 4y.
When multiplying
polynomials, remember the distributive law and multiply every term in the
first polynomial by every term in the second polynomial (FOIL). For example,
(x + 4)(2x –
1) = x(2x) + x(-1) + 4(2x) + 4(-1)
= 2x2+ (-x) + 8x + (-4)
= 2x2 + 7x – 4
There are three
expressions that you need to commit to memory for the GRE in both their
factored and unfactored forms. They are:
1. x2 – y2 = (x + y)(x – y)
x2 – y2 is the unfactored form; (x + y)(x –
y) is the factored form.
2. x2 + 2xy + y2 = (x + y)(x + y) = (x + y)2
x2 + 2xy + y2 is the unfactored form; (x + y)(x + y) and
(x + y)2 are equivalent
factored forms.
3. x2 – 2xy + y2 = (x – y)(x – y) = (x – y)2
x2 – 2xy + y2 is the unfactored form; (x –
y)(x – y) and (x – y)2 are equivalent
factored forms.
Here is an example
of a problem in which you need to factor and know one of these
expressions. Example: Simplify the following expression:
4x2 – 4
x – 1
You should immediately
recognize that you need to factor the numerator which results in the
expression 4(x2 – 1) for the numerator. Next, you should recognize that x2 – 1 is of the form x2
– y2 and therefore can be written in its factored form (x – 1)(x + 1). Hence,
the original fraction
can be written as
4(x + 1)(x –
1)
(x – 1)
Now you should
recognize that the fraction can be reduced by canceling out the common
factor (x – 1) in the numerator and the denominator. The final simplified, factored form is 4(x
+
1).
Whenever you see
a complicated-looking algebraic expression, simplify it if possible by
combining similar terms, i.e. terms with like bases and like exponents. For example,
(4x2
+ 4x + 2) + (3 – 7x) – (5 – 3x)
= 4x2 + 4x + 2 + 3 – 7x – 5 + 3x
= 4x2 + (4x – 7x + 3x) + (2 + 3 – 5)
= 4x2
When given two simultaneous
equations on the GRE and asked to solve them, don’t solve
them using the techniques you learned in algebra class. Instead, look for shortcuts which usually
involve adding or subtracting the two equations. And you’ll never have to worry about solving
a
system with more than two equations.
Let’s see
an example. If 5x + 4y = 6 and 4x + 3y = 5, then x + y =? Instead of using
substitution or elimination like you learned in algebra, just add or subtract the two equations.
First, let’s add them and see what we get. Remember, we are looking for x + y, so we really
don’t need to know what x and y are individually.
5x + 4y = 6
+ 4x + 3y = 5
9x + 7y = 11
Obviously, that
didn’t help us much. So let’s subtract them.
5x + 4y = 6
-(4x + 3y = 5)
x + y = 1
Since we’re
looking for the value of the expression x + y, the answer would be 1. We never
even had to find out the values of x and y. Whenever you encounter simultaneous equations, try
adding or subtracting them, factoring something out, or multiplying by something. You will never
need to use the methods you learned in your algebra classes to solve simultaneous equations on
the GRE.
The following is
an example of the type of problem on the GRE that involves factoring and the
distributive law. Example: If y + 3 = 2x, then 3y – 6x =
(a) –9 (b) –3 (c) 0 (d) 3 (e) 9
Instead of solving
for y and plugging –3 + 2x in for y in the second expression, just factor out a
three in the second expression: 3(y – 2x). This is just the distributive law working in reverse.
Notice that the first equation can be written y – 2x = -3 (why?). By substituting –3 in
for y –
2x, we get 3(-3) = -9, so the answer is (a). Always be on the lookout for chances to factor and
use the distributive laws.
The GRE loves equations
set equal to zero because of the unique properties of 0. One of the
most important properties of 0, which was mentioned in Review #1, is the fact that the product
of anything and 0 is 0. Hence, if a product is equal to 0, one of the factors in that product must
be 0. In other words, if ab = 0, then either a or b must be 0 or both are 0. This fact can be
used to solve some equations on the GRE.
Here’s an example. What
are all the values of y for which y(y + 5) = 0? In order for the
product y(y +5) to equal 0, either y must be 0 or y + 5 must be 0 or both of them must be 0. In
order for y +5 to be 0, y would have to be –5. Hence, the values of y for which y(y + 5) is 0
are y = 0 and y = -5.
In an equation,
one expression is equal to another. In an inequality, one expression is not
equal to another expression. (See Review #1, page 3, for a list of symbols and their meanings.)
However, inequalities are solved just like equations. You can factor, unfactor, simplify,
multiply/divide both sides by a constant, add/subtract terms from both sides, etc. The one
primary difference is that if you multiply or divide both sides of an inequality by a
negative number, you must reverse the direction of the inequality symbol. It’s easy to
see why. For instance, we know that 2 < 4. Multiplying both sides by –2 results in –4
on the
left side and –8 on the right side. Clearly, -4 is greater than –8, so the inequality symbol
must
be reversed: -4 > -8.
From your algebra
classes, you probably remember hearing about functions even if didn’t you
understand them. Usually, the symbol f(x), read "f of x", was used to represent a function.
The
GRE contains function problems, but instead of using the f(x), they are disguised by funny-
looking symbols such as #, *, @, etc. If you remember how functions work, just think of
functions when you work these problems and perform the operations. However, you can still
work these problems even if you don’t remember functions. Think of a funny-looking symbol
as representing a set of operations or instructions. Here’s an example: If x @ y = (x –
y )/2,
what is the value of 3 @ 5? To find the answer, just substitute 3 in for x and 5 in for y. Since x
@ y = (x – y)/2, then 3 @ 5 = (3 – 5)/2 = -1.
The GRE also contains
your favorite type of problems: word problems. You have to learn
how to translate them into mathematical equations. The following is list of words found in word
problems and their mathematical translations. These are the same translations that are used in
percentage problems (see Review #4).
Word Symbol
is =
of, times, product
what (or the unknown
value) any variable
more, sum +
less, difference
-
ratio, quotient
The following formulas frequently
appear in GRE word problems, so you should commit them
to memory:
1. distance = (rate)(time)
This formula can also be expressed as rate = distance/time or time =
distance/rate.
2. total price =
(number of items)(cost per item)
3. sale price = (original price) – (% discount)(original price)
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