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Intermediate Algebra
Chapter 7
Section 7.1
• Opposite of squaring a number is taking the
•
•
square root of a number.
A number b is a square root of a number a if
b2 = a.
In order to find a square root of a, you need
a # that, when squared, equals a.
The principal (nonnegative) square root is
noted as
a
The negative square root is noted as
 a
• Radical expression is an expression
•
•
containing a radical sign.
Radicand is the expression under a radical
sign.
Note that if the radicand of a square root is
a negative number, the radical is NOT a real
number.
Example
49 
7
5
25

16
4
 4  2
• Square roots of perfect square radicands
•
•
simplify to rational numbers (numbers that
can be written as a quotient of integers).
Square roots of numbers that are not perfect
squares (like 7, 10, etc.) are irrational
numbers.
IF REQUESTED, you can find a decimal
approximation for these irrational numbers.
• Otherwise, leave them in radical form. Do not
convert to an approximation unless requested to
do so.
Radicands might also contain variables and
powers of variables.
Example
Simplify. Assume that all variables represent
positive numbers.
64x10  8x 5
2 20
25a b
 5ab
10
The cube root of a real number a
3
a  b only if b 3  a
Note: a is not restricted to non-negative
numbers for cubes. The cube root of a
negative number is a negative number.
Example
3
3
27  3
 8x   2x
6
3
4
a
64
a
3
  3
9
b
b
2
Other roots can be found, as well.
The nth root of a is defined as
n
a  b only if b n  a
If the index, n, is even, the root is NOT a
real number when a is negative.
If the index is odd, the root will be a real
number when a is negative.
Example
Simplify the following. Assume that all
variables represent positive numbers.
4
2
2x
16x 
8
2a
32a
 2a 
  15     3   3
b
b
 b 
5
5
n
If the index of the root a is even, then the
notation represents a positive number.
But we may not know whether the variable a is
a positive or negative value.
Since the positive square root must indeed be
positive, we might have to use absolute value
signs to guarantee the answer is positive.
If n is an even positive integer, then
n
an  a
If n is an odd positive integer, then
n
a a
n
Example
Simplify the following.
2 20
25a b
 5ab
10
 5 a b10
If we know for sure that the variables represent
positive numbers, we can write our result
without the absolute value sign.
2 20
25a b
 5ab
10
Example
Simplify the following.
3
4
a
64
a
3
  3
9
b
b
Since the index is odd, we don’t have to force
the negative root to be a negative number.
If a or b is negative (and thus changes the sign
of the answer), that’s okay.
Since every value of x that is substituted into the
equation
yn x
produces a unique value of y, the root relation
actually represents a function.
The domain of the root function when the index
is even, is all nonnegative numbers.
The domain of the root function when the index
is odd, is the set of all real numbers.
We have previously worked with graphing basic
forms of functions so that you have some familiarity
with their general shape.
You should have a basic familiarity with root
functions, as well.
Example
x
y
6
6
4
2
2
1
2
1
0
0
Graph y  x
y
(6, 6 )
(4,
2)
(2, 2 )
(1, 1)
x
(0, 0)
Example
x
y
8
2
4
3
y
4
1
1
0
0
-1
-1
-4
 4
-2
-8
Graph y  3 x
3
3
(4, 4 )
(8, 2)
(1, 1)
(-1, -1)
(-8, -2)
(-4,  3 4)
(0, 0)
x