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Math 52
8.1 "Introduction to Radical Expressions"
Objectives:
*
Find the principal square roots and their opposites.
*
Solve applied problems involving square roots.
*
Identify radicands of radical expressions.
*
Identify whether a radical expression represents a real number.
*
Simplify a radical expression with a perfect square radicand.
Square Roots
When we raise a number to the second power, we have squared the number. Sometimes we may need to …nd the number
that was squared. We call this process …nding a square root of a number.
De…nition:
"Square Root"
The number c is a square root of a if
:
Every positive number has two square roots. For example, the square roots of 25 are 5 and
2
( 5) = 25: The positive square root is also called the principal square root. The symbol
5 because 52 = 25 and
is called a radical
(or square root) symbol. The radical symbol represents only the principal square root. To name the negative square root
of a number, we use
:
The number 0 has only one square root, 0:
Example 1: (Finding square roots)
Find the square roots of the following numbers and tell which is the principal square root:
a) 36
b) 144
c) 225
Example 2: (Simplifying square roots)
Simplify:
p
16
a)
b)
p
169
c)
p
441
Radicands and Radical Expressions
De…nition:
"Radical Expression" and "Radicand"
A radical expression is an expression written under a radical.
The expression under the radical is called radicand.
Example 3: (Identifying radicands)
Identify the radicand in each expression:
p
p
a)
105
b) 2 x + 2
p
c) 2 x + 2
Page: 1
d)
r
a b
a+b
Notes by Bibiana Lopez
Introductory Algebra by Marvin L. Bittinger
8.1
Expressions that Are Meaningful as Real Numbers
Excluding Negative Radicands:
kRadical expressions with negative radicands do not represent real numbers.k
Example 4:
Determine whether the expression represents a real number. Write "yes" or "no."
p
p
p
a)
25
b)
25
c)
36
p
d)
36
Perfect-Square Radicands
Principal Square Root of A2 :
For any real number A,
:
(That is, for any real number A, the principal square root of A2 is the absolute value of A)
Example 5: (Simplifying radicals of perfect-square radicands)
Simplify
q (Assume that the expressions
q under the radicals represent
q any real number):
2
2
2
a)
( 13)
b)
(7w)
c)
(xy)
d)
p
x2 + 8x + 16
Fortunately, in many cases, it can be assumed that radicands that are variable expressions do not represent the square
of a negative number. When this assumption is made, the need for the absolute-value symbols disappears. Then for x
p
x2 = x; since x is nonnegative.
Principal Square Root of A2 :
For any nonnegative real number A,
0;
:
(That is, for any nonnegative real number A, the principal square root of A2 is A)
Example 6: (Simplifying radicals of perfect-square radicands)
Simplify (Assume that radicands do not represent the square of
ra negative number):
p
p
1 2
b)
25y 2
c)
t
a) 9x2
4
Page: 2
d)
p
x2 + 2x + 1
Notes by Bibiana Lopez