Sat math Essentials

a.
2
b.
3
c.
6
d.
9
e.
18
Answer
e.
6
n
means 6 
n
, so 6
n
6 
(8
5) 
6 
3 
18.
Like Terms
A 
variable
is a letter that represents an unknown number. Variables are used in equations, formulas, and math-
ematical rules.
A number placed next to a variable is the 
coefficient
of the variable:
9
d
9 is the coefficient to the variable 
d
.
12
xy
12 is the coefficient to both variables, 
x
and 
y
.
If two or more terms contain exactly the same variables, they are considered 
like terms
:
4
x
, 7
x,
24
x,
and 156
x
are all like terms.
8
ab
, 10
ab,
45
ab,
and 217
ab
are all like terms.
Variables with different exponents are 
not
like terms. For example, 5
x
3
y
and 2
xy
3
are not like terms. In the
first term, the 
x
is cubed, and in the second term, it is the 
y
that is cubed.
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3 9

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You can 
combine like terms
by grouping like terms together using mathematical operations:
3
x
9
x
12
x
17
a
6
a
11
a
Practice Question
4
x
2
y
5
y
7
xy
8
x
9
xy
6
y
3
xy
2
Which of the following is equal to the expression above?
a.
4
x
2
y
3
xy
2
16
xy
8
x
11
y
b.
7
x
2
y
16
xy
8
x
11
y
c.
7
x
2
y
2
16
xy
8
x
11
y
d.
4
x
2
y
3
xy
2
35
xy
e.
23
x
4
y
4
8
x
11
y
Answer
a.
Only like terms can be combined in an expression. 7
xy
and 9
xy
are like terms because they share the
same variables. They combine to 16
xy
. 5
y
and 6y are also like terms. They combine to 11
y
. 4
x
2
y
and
3
xy
2
are not like terms because their variables have different exponents. In one term, the 
x
is squared,
and in the other, it’s not. Also, in one term, the 
y
is squared and in the other it’s not. Variables must
have the exact same exponents to be considered like terms.
P r o p e r t i e s o f A d d i t i o n a n d M u l t i p l i c a t i o n
■
Commutative Property of Addition
. When using addition, the order of the addends does not affect the
sum:
a
b
b
a
7 
3 
3 
7
■
Commutative Property of Multiplication
. When using multiplication, the order of the factors does not
affect the product:
a
b
b
a
6 
4 
4 
6
■
Associative Property of Addition.
When adding three or more addends, the grouping of the addends does
not affect the sum.
a
(
b
c
) 
(
a
b
) 
c
4 
(5 
6) 
(4 
5) 
6
■
Associative Property of Multiplication.
When multiplying three or more factors, the grouping of the fac-
tors does not affect the product.
5(
ab
) 
(5
a
)
b
(7 
8) 
9 
7 
(8 
9)
■
Distributive Property
. When multiplying a sum (or a difference) by a third number, you can multiply each
of the first two numbers by the third number and then add (or subtract) the products.
7(
a
b
) 
7
a
7
b
9(
a
b
) 
9
a
9
b
3(4 
5) 
12 
15
2(3 
4) 
6 
8
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Practice Question 
Which equation illustrates the commutative property of multiplication?
a.
7(
8
9
1
3
0
) 
(7 
8
9
) 
(7 
1
3
0
)
b.
(4.5 
0.32) 
9 
9 
(4.5 
0.32)
c.
12(0.65 
9.3) 
(12 
0.65) 
(12 
9.3)
d.
(9.04 
1.7) 
2.2 
9.04 
(1.7 
2.2)
e.
5 
(
3
7
4
9
) 
(5 
3
7
) 
4
9
Answer
b.
Answer choices 
a
and 
c
show the distributive property. Answer choices 
d
and 
e
show the associative
property. Answer choice 
b
is correct because it represents that you can change the order of the terms
you are multiplying without affecting the product.
Order of Operations
You must follow a specific order when calculating multiple operations:
P
arentheses: First, perform all operations within parentheses.
E
xponents: Next evaluate exponents.
M
ultiply/
D
ivide: Then work from left to right in your multiplication and division.
A
dd/
S
ubtract: Last, work from left to right in your addition and subtraction.
You can remember the correct order using the acronym 
PEMDAS
or the mnemonic 
P
lease 
E
xcuse 
M
y 
D
ear
A
unt 
S
ally
.
Example
8 
4 
(3 
1)
2
8 
4 
(4)
2
P
arentheses
8 
4 
16
E
xponents
8 
64
M
ultiplication (and 
D
ivision)
72
A
ddition (and 
S
ubtraction)
Practice Question
3 
(49 
16) 
5 
(2 
3
2
) 
(6 
4)
2
What is the value of the expression above?
a.
146
b.
150
c.
164
d.
220
e.
259
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Answer
b.
Following the order of operations, the expression should be simplified as follows:
3
(49 
16) 
5
3 (2 
3
2
) 
(6 
4)
2
3
(33) 
5
(2 
9) 
(2)
2
3
(33) 
5
(11) 
4
[3
(33)] 
[5
(11)] 
4
99 
55 
4
150
P o w e r s a n d R o o t s
Exponents
An 
exponent
tells you how many times a number, the 
base,
is a factor in the product.
3
5
3 
3 
3 
3 
3 
243
3 is the 
base
. 5 is the 
exponent
.
Exponents can also be used with variables. You can substitute for the variables when values are provided.
b
n
The “
b
” represents a number that will be a factor to itself “
n
” times.
If
b
4 and 
n
3, then 
b
n
4
3
4 
4 
4 
64.
Practice Question 
Which of the following is equivalent to 7
8
?
a.
7 
7 
7 
7 
7 
7
b.
7 
7 
7 
7 
7 
7 
7 
c.
8 
8 
8 
8 
8 
8 
8
d.
7 
7 
7 
7 
7 
7 
7 
7
e.
7 
8 
7 
8
Answer
d.
7 is the base. 8 is the exponent. Therefore, 7 is multiplied 8 times.
Laws of Exponents
■
Any base to the zero power equals 1.
(12
xy
)
0
1
80
0
1
8,345,832
0
1
■
When multiplying identical bases, keep the same base and add the exponents:
b
m
b
n
b
m
n
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Examples
9
5
9
6
9
5 
6
9
11
a
2
a
3
a
5
a
2 
3 
5
a
10
■
When dividing identical bases, keep the same base and subtract the exponents:
b
m
b
n
b
m
n
b
b
m
n
b
m 
n
Examples
6
5
6
3
6
5 
3
6
2
a
a
9
4
a
9 
4
a
5
■
If an exponent appears outside of parentheses, multiply any exponents inside the parentheses by the expo-
nent outside the parentheses.
(
b
m
)
n
b
m
n
Examples
(4
3
)
8
4
3 
8
4
24
(
j
4
k
2
)
3
j
4 
3
k
2 
3
j
12
k
6
Practice Question 
Which of the following is equivalent to 6
12
?
a.
(6
6
)
6
b.
6
2
6
5
6
5
c.
6
3
6
2
6
7
d.
1
3
8
3
15
e.
6
6
4
3
Answer
c.
Answer choice 
a
is incorrect because (6
6
)
6
6
36
. Answer choice 
b
is incorrect because exponents don’t
combine in addition problems. Answer choice 
d
is incorrect because 
b
b
m
n
b
m
n
applies only when the
base in the numerator and denominator are the same. Answer choice 
e
is incorrect because you must
subtract the exponents in a division problem, not multiply them. Answer choice 
c
is correct: 6
3
6
2 
6
7
6
3 
2 
7
6
12
.
S q u a r e s a n d S q u a r e R o o t s
The 
square
of a number is the product of a number and itself. For example, the number 25 is the 
square
of the
number 5 because 5 
5 
25. The square of a number is represented by the number raised to a power of 2:
a
2
a
a
5
2
5 
5 
25
The 
square root
of a number is one of the equal factors whose product is the square. For example, 5 is the
square root of the number 25 because 5 
5 
25. The symbol for square root is 
. This symbol is called the