Transcript Section 1.4 a

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Hawkes Learning Systems:
College Algebra
Section 1.4a: Properties of Radicals
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Objectives
o Roots and radical notation.
o Simplifying radical expressions.
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Roots and Radical Notation
nth Roots and Radical Notation
Case 1: n is an even natural number. If a is a non-negative
real number and n is an even natural number, n a is the
non-negative real number b with the property that b n  a.
n
n
That is a  b  a  b .In this case, note that
 
n
a
n
 a and n a n  a.
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Roots and Radical Notation
nth Roots and Radical Notation (cont.)
Case 2: n is an odd natural number. If a is any real
number and n is an odd natural number, n a is the real
number b (whose sign will be the same as the sign of a)
with the property that b n  a. Again,
n
n
n
n
a  b  a  b , a  a and n a n  a.
 
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Roots and Radical Notation
nth Roots and Radical Notation (cont.)
The expression n a expresses the nth root of a in radical
notation. The natural number n is called the index, a is the
radicand and
is called the radical sign. By convention,
2
is usually written as
. Radical Sign
Index
n
a
Radicand
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Roots and Radical Notation
Note:
o When n is even, n a is defined only when a is nonnegative.
o When n is odd, n a is defined for all real numbers a.
o We prevent any ambiguity in the meaning of n a
when n is even and a is non-negative by defining n a
to be the non-negative number whose nͭ ͪ power is a.
Ex: 4  2, NOT  2.
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Example 1: Roots and Radical Notation
Simplify the following radicals.
a.
3
27  3
b.
4
81
because  3  27.
3
is not a real number, as no real number raised to the fourth power is -81.
0 0
Note:
n
0  0  0n  0 for any natural number n.
d. 1  1
Note:
n
1  1  1n  1 for any natural number n.
c.
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Example 2: Roots and Radical Notation
Simplify the following radicals.
a.
3
5
125


6
216
b. 3  3  
c.
6
 5 
6
3
125
 5
because     
216
 6
because
 6 15625  5
  
3
 (1)3 3   3
because 56  15625  (5)6
In general, if n is an even natural number, n a n  a for any real number a.
Remember, though, that n an  a if n is an odd natural number.
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Simplifying Radical Expressions
Simplified Radical Form
A radical expression is in simplified form when:
( Use as reference, n a )
1. The radicand contains no factor with an exponent greater than or equal
to the index of the radical (exponents in a  n) .
x
2. The radicand contains no fractions (
).
yx
3. The denominator contains no radical (
).
a
4. The greatest common factor of the index and any exponent occurring in
the radicand is 1. That is, the index and any exponent in the radicand have
no common factor other than 1 ( GCF(any exponent in a, n)=1 ).
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Simplifying Radical Expressions
In the following properties, a and b may be taken to represent
constant variable, or more complicated algebraic expressions.
The letters n and m represent natural numbers.
Property
1. n ab  n a  n b
a

b
2.
n
3.
m n
4
n
a
n
b
a  mn a
3
4
Example
4 x8 y 3  4 4  4 x8  4 y 3  4 4  x 2  4 y 3  x 2  4 4 y 3
x3

8
3
x3 x

3
8 2
256 
4 2
256  8 256  2
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Example 3: Simplifying Radical Expressions
Simplify the following radical expressions:
a. 54x y 
3
10
5
3
 3 
3
 2 x

3 3
 3 x 3 y 3 2  x  y 2
b.
12z
10
 2  3  z
2
 2z5
3
 x  y3  y 2
Note that since the index is 3,
we look for all of the perfect
cubes in the radicand.

5 2
n
a n  a if n is even.
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Example 3: Simplifying Radical Expressions (cont.)
Simplify the following radical expressions:
c.
3 3
2
108x 2
3

4

x
3

6
y
2 3
3
y 
3 3 4x 2

y2
All perfect cubes have been brought
out from under the radical, and the
denominator has been rationalized.
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Simplifying Radical Expressions
Caution!
One common error is to rewrite
a  b as a  b
These two equations are not equal! To convince
yourself of this, observe the following inequality:
 9  16 
9  16
 3 4
 25
5
7
75
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Simplifying Radical Expressions
Rationalizing Denominators
Case 1: Denominator is a single term containing a root.
If the denominator is a single term containing a factor of n a m
we will take advantage of the fact that
n
a m  n a n m  n a m  a n m  n a n
and n a n is a or |a|, depending on whether n is odd or
even.
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Simplifying Radical Expressions
Rationalizing Denominators
Case 1: Denominator is a single term containing a root.
(cont.)
Of course, we cannot multiply the denominator by a factor
of n a n m without multiplying the numerator by the same
factor, as this would change the expression. So we must
multiply the fraction by n a n m
n
a
n m
.
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Simplifying Radical Expressions
Rationalizing Denominators
Case 2: Denominator consists of two terms, one or both of which
are square roots.
Let A + B represent the denominator of the fraction under
consideration, where at least one of A and B is a square root term.
We will take advantage of the fact that
 A  B  A  B   A A  B   B  A  B   A2  AB  AB  B2  A2  B2
Note that the exponents of 2 in the end result negate the square
root (or roots) initially in the denominator.
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Simplifying Radical Expressions
Rationalizing Denominators
Case 2: Denominator consists of two terms, one or both of which
are square roots.
(cont.)
Once again, remember that we cannot multiply the denominator
by A – B unless we multiply the numerator by this same factor.
A– B
.
Thus, multiply the fraction by
A– B
The factor A – B is called the conjugate radical expression of A + B.
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Example 4: Simplifying Radical Expressions
Simplify the following radical expression:
2 x

4y
4
3

3
2 x  x
3 4y
 x 3 2x
3

2
3

2 y
 x 3 4 xy 2
3
23 y 3
 x 3 4 xy 2

2y
3
2  y2
3
2 y
2
First, simplify the numerator and
denominator. Since the index is three,
we are looking for perfect cubes. Here,
the perfect cubes are colored green.
Next, determine what to multiply the
denominator by in order to eliminate
the radical. Since the index is three, we
want 2 and y to have exponents of
three. Remember to multiply the
numerator by the same factor.
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Example 5: Simplifying Radical Expressions
Simplify the following radical expression:
3
8 5
Find the conjugate radical of the


8  5 denominator to eliminate the radicals.
3



 Then, multiply both the numerator and
 
 8 5  8 5



8 5

85
3 8 5

3



3
8 5

the denominator by it.
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Example 6: Simplifying Radical Expressions
Simplify the following radical expression:
 7x
7 x
  7x   7  x 


 
 7 x   7 x 
7 7 x  7 x 2

49  x 2
7 7 x  x 7

49  x
Again, find the conjugate radical of the
denominator to eliminate the radicals.
Then, multiply both the numerator and
the denominator by it.
Since the index is two, we are looking for
perfect squares. In this example, the perfect
squares are colored green.
Note: the original expression is not real if x is
negative. Since x must be positive, when we
pull x out of 7x2 we do not need to write
x , but simply x.
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Example 7: Simplifying Radical Expressions
Rationalize the numerator in the following radical
expression:
Find the conjugate radical of the numerator to
8 x  2 y 8 x  2 y eliminate the radicals in the numerator. Then,

4x  y
8 x  2 y multiply both the numerator and the
denominator by it and simplify as usual.
8x  2 y

 4 x  y  8x  2 y


2 4x  y 
 4x  y  
2

8x  2 y
8x  2 y

