HAWKES LEARNING SYSTEMS
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Copyright © 2011 Hawkes Learning Systems.
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Hawkes Learning Systems:
Developmental Mathematics
Section 14.1: Roots and Radicals
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Perfect Squares and Square Roots
Radical Terminology
The symbol
radical sign.
is called a
The number under the
radical sign is called the
radicand.
The complete expression,
such as 64 , is called a
radical or radical
expression.
Square Root
If a is a nonnegative real
number, then
a is the principal square
root of a,
and
 a is the negative square
root of a.
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
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math courseware specialists
Perfect Squares and Square Roots
Notes
Square roots of negative numbers are not real
numbers. For example, 4 is not a real number.
There is no real number whose square is 4. Numbers
of this type are discussed in Appendices A.8 and A.9.
HAWKES LEARNING SYSTEMS
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Copyright © 2011 Hawkes Learning Systems.
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Evaluating Square Roots
o 64 has two square roots, one positive and one
negative. The
sign is understood to represent
the positive square root (or the principal square
root) and  represents the negative square root.
Therefore, we have and
64 = 8 and
 64 = 8
o Because 112 = 121, we have 121 = 11 and  121 = 11.
o Because 02 = 0, 02 = 0.
o 25 is not a real number.
HAWKES LEARNING SYSTEMS
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Copyright © 2011 Hawkes Learning Systems.
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Evaluating Square Roots
2
16 4
4  16

o Because   = , we know that
= .
25 5
 5  25
o  0.0009 = 0.03 because
 0.03
2
= 0.0009.
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
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math courseware specialists
Cube Roots
Cube Roots
If a is a real number, then
is the cube root of a.
Notes
In the cube root expression 3 a the number 3 is called
the index. In a square root expression such as a the
index is understood to be 2 and is not written.
Expressions with square roots and cube roots (as well
as other roots) are called radical expressions.
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Evaluating Cube Roots
If a is a real number, then
Because
23 = 8,
Because
 6  = 216,
3
3
a is the cube root of a.
3
216 = 6.
8 = 2.
3
Note that the cube root of a negative number is a real number and is
negative.
3
Because
1 1
  = ,
 3  27
3
1 1
= .
27 3
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Evaluating Radical Expressions
with a Calculator