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College Prep 8.1
Radical Expressions and Graphs
Square Root: The number c is a square root of a if c 2  a.
2
9 has –3 and 3 as square roots because  3  9 and 32  9.
The principal square root of a nonnegative number is its nonnegative square root. Whenever we talk about
“the square root” of a number, we mean its principal square root. The square root of a negative number is not
a real number.
The symbol
is called a radical sign and is used to indicate the principal square root of the number over
which it appears. The number under the radical sign is called the radicand.
If a is a nonnegative number, then
of a.
a is the principal square root of a, and  a is the negative square root
Perfect squares are numbers that are the squares of rational numbers. Examples: 1, 4, 9, 81,
Examples: Simplify each of the following:
36
a) 121
b)
49
c)  81
1
36
,
16
25
, etc.
0.36
d)
Simplifying a 2 : For any real number a, a 2  a .
Remember: The principal square root must be positive. When a radicand is the square of a variable expression,
2
like  x  5  or 36t 2 , we must use absolute-value signs unless we know that the expression being squared is
nonnegative.
Examples: Simplify each expression. Assume that the variable can represent any real number.
 x  5
a)
2
x2  6 x  9
b)
c)
y4
t10
d)
Examples: Simplify each expression. Assume that no radicands were formed by squaring negative quantities.
y6
a)
b)
4 x2  12 x  9
Cube Root: The number b is the cube root of a if and only if b3  a.
3
a denotes the cube root of a.
Examples: Simplify each expression
a)
3
27
b)
3
27
c)  3 27
d)
3
8m3
Odd and Even nth roots
In the expression n a , n is called the index and we are finding the nth root of a.
If the index is even, such as
If the index is odd, such as
3
or
4
, the radicand must be nonnegative for the root to be a real number.
or
5
, the radicand can be either positive or negative.
Simplifying nth roots
n
n
a
a
Positive
Positive
Negative
Not a real number
Positive
Positive
Negative
Negative
Even
n
an
a
a
Odd
Examples: Simplify each expression, if possible. Assume that variables can represent any real number.
a) 5 243
b) 5 243
c)  5 243
d)  5 243
e)
9
x9
i)  4 16
f)
7
 2 x  3
j) 4 16x 4
7
g) 4 16
h)
x8
l)
k)
4
16
4
6
 m  3
6
Domain: The domain of an even root is all values of the variable that make the radicand nonnegative. The
domain of an odd root is .
Examples: Determine the domain of each function described.
a) f  x   x  3
b) 3 y  7
c) g  x   4 2 x  5
d)
7
3  2z
Examples: Graph each function by creating a table of values. Give the domain and range.
a) f  x   x  1
b) g  x   3 x  4
HW 8.1 Problems: 13,15,18,21,24,27,30,32,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78