RADICALS
COMMON MISTAKES
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10/20/2009
Radicals-Notation, Definition, and
Simplifying
How to Understand the
Definition and Notation
Notation:
a
n= root,
= radical, a= radicand.
 Square root, n=2, but the two is NOT written
 (i.e. 16 )
3
 Cube root, n=3, (i.e.
8 ).
 Definition and Simplifying:
n
a requires a to be factored into repeating
factors and for n repeating factors, one factor
can be “pulled through” the radical:
Examples: n=2 requires the factor to repeat twice
n

Common Mistakes
 Not correctly factoring into
simplified form.

Incorrect:
48 = 4 • 12
= 4 • 12
= 2 • 12
16 = 4 • 4 = 4
n=3 requiring the factor to repeat
three times
3
3
8 = 2•2•2 = 2

Correct:
48 = 16 • 3
= 16 • 3
= 4• 3
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Complete Manual: ..\Radical Review.docx
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10/20/2009
Radicals-Negative Radicals
How to Evaluate Negative
Radicals
Since a (where n is a
positive integer,) implies only
the positive roots, there are
times when the radicand, a ,
can be negative.
If the radicand, a, is negative,
then it is defined only if n is
a positive ODD integer.
n


Common Mistakes
 Trying to evaluate negative
radicands when n is even.
 Confusing − 2 and − 2 .
 Forgetting the negative sign
when evaluating radicals
with POSITIVE ODD
ROOTS.
3
− 8.
 Example: Simply
Solution:
Since n=3 is a positive odd
integer, a=-8 is permissible.
So, − 8 = − 2 • −2 • −2 = −2 .
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10/20/2009
Radicals-Multiplication/Division
How to Multiply/Divide
Radicals
Multiplication

a • b = ab
Division

n
a
a
=
b
b
n
n
Note: n must be the same
and a and b must be defined
such that n yields a Real
solution.
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Common Mistakes


Not combining or separating radical
expressions when simplifying
expressions.
Example 1: Simplify: 2 • 8 .
Solution:
= 2•8
= 16 = 4
Example 2: Simplify:
Solution:
=
27
= 9=3
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10/20/2009
27
.
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Radicals-Addition/Subtraction
How to Add/Subtract
Radicals
Common Mistakes

Note: Radicals can only
be added/subtracted
together if they have the
and same root with the
same radicand.
Sometimes, radicals must
be simplified before they
can be combined.

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Complete Manual: ..\Radical Review.docx
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
Not reducing radicals to their
SIMPLIFIED form before trying to add
or subtract the radicals.
Example 1: Simplify.
3
16 + 2 − 14
3
3
Solution:
First, 3 16 = 3 8 • 2 = 3 8 • 3
which then substitutes to
become …
2,
= 2 2 + 1 2 − 14
3
3
= 3 2 − 14
3
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10/20/2009
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Radicals- Rationalizing
How to Rationalize
Radicals


A radical is considered to be in proper
form if there is no radical in the
denominator( the bottom number in a
fraction).
To put a radical expression in proper
form is called rationalizing. It involves
multiplying the expression by a clever
form of 1. It is the radical in the
denominator that will indicate what that
form will be.
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Complete Manual: ..\Radical Review.docx
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Common Mistakes


Choosing the incorrect clever
form of 1.
4 .
Simplify
7

Incorrect:

Correct:
4 7
28
• =
7 7 7 7
4
7 4 7
•
=
7
7
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10/20/2009
Radicals- Conjugating Radical Expressions
How to Conjugate Radical
Expression
Given a + b , the conjugate
is a − b or visa versa.
Conjugates are multiplied together
which cancels out the radical
because ( x + y )( x − y ) = x − y
shows that the middle terms “fall
out”.


2
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Common Mistakes



Using the “wrong” conjugate.
Incorrectly multiplying the terms
together.
Simplify by rationalizing the
3
denominator:
4+ 2
.


Incorrect: Conjugate is− 4 + 2
Correct: Conjugate is 4 − 2
So… 3 • 4 − 2
4+ 2 4− 2
3(4 − 2 ) 3(4 − 2 )
=
=
14
16 − 2
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Complete Manual: ..\Radical Review.docx
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10/20/2009
Radicals- Solving Radical Equations
How to Solve Radical
Equations
When solving equations involving
radicals, the first idea is to isolate
the radical onto only one side of
the equation before attempting to
solve the equation.
Then, raise both sides to the
power that is the reciprocal of the
root. Now, the variable is to the
first power and is in simplified
form.


Common Mistakes




Not noticing the value of the root
problem, resulting in solving for
the wrong power.
Not repeating the process of
solving until the variable is to the
first power.
Example: Solve 3 − x = −6 for x.
Solution:
− x = −9
x=9
( x) = 9
x = 81
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Complete Manual: ..\Radical Review.docx
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10/20/2009
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