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Slide 10- 1
7.3
Multiplying
Radical
Expressions
■
■
■
Multiplying Radical Expressions
Simplifying by Factoring
Multiplying and Simplifying
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Multiplying Radical Expressions
Note that
4  9  2  3  6.
4  9  36  6.
This example suggests the following.
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Slide 7- 3
The Product Rule for Radicals
For any real numbers n a and n b ,
n a  n b  n a  b.
(The product of two nth roots is the nth root of the
product of the two radicands.)
Rational exponents can be used to derive this
rule:
n a  n b  a1/ n  b1/ n  a  b 1/ n  n a  b.
 
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Slide 7- 4
Example
Multiply.
a)
5 6
Solution
a)
5  6  5  6  30.
b) 3 7  3 9
b) 3 7  3 9  3 7  9  3 63
x
5
c) 4  4
3 z
x
5
x
5
5
x
c) 4  4  4   4
3 z
3 z
3z
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Slide 7- 5
CAUTION!
The product rule for radicals applies only
when radicals have the same index:
n a  m b  nm a  b .
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Slide 7- 6
Simplifying by Factoring
An integer p is a perfect square if there
exists a rational number q for which q2 = p.
We say that p is a perfect cube if q3 = p for
some rational number q. In general, p is the
perfect nth power if qn = p for some rational
number q.
The product rule allows us to simplify
when a or b is a perfect nth power.
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n ab
Slide 7- 7
Using The Product Rule to
Simplify
n ab  n a  n b
( n a and n b must both be real numbers.)
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Slide 7- 8
To Simplify a Radical Expression with
Index n by Factoring
1. Express the radicand as a product in which one
factor is the largest perfect nth power possible.
2. Rewrite the expression as the nth root of each
factor.
3. Simplify the expression containing the perfect
nth power.
4. Simplification is complete when no radicand
has a factor that is a perfect nth power.
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Slide 7- 9
It is often safe to assume that a radicand
does not represent a negative number raised
to an even power. We will henceforth make
this assumption ––unless functions are
involved –– and discontinue use of absolutevalue notation when taking even roots.
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Slide 7- 10
Example Simplify by factoring:
a)
300
b)
8m 4 n
c)
3
54 s
4
Solution
a)
300  100  3
100 is the largest perfectsquare factor of 300.
 100  3
 10 3
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Slide 7- 11
Solution continued
8m4n  4  2  m4  n
b)
 4m 4  2n
 2m
c)
3
2
2n
3
54s 4  27  2  s3  s
27s3 is the largest perfect
third-power factor.
3
 27 s3  3 2s
 3s 3 2s
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Slide 7- 12
Example
If f  x   2x2  12x  18, find a simplified form for f(x).
Solution
f  x 
2 x2  12 x  18
 2  x2  6 x  9

 x  3

 x  3  2
2
2
 x3 2
2
Factoring into two radicals.
Taking the square root.
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Slide 7- 13
continued
We can check by graphing
y1  2 x2  12 x  18 and y2  x  3 2.
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Slide 7- 14
To simplify an nth root, identify factors in
the radicand with exponents that are
multiples of n.
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Slide 7- 15
Multiplying and Simplifying
We have used the product rule for radicals
to find products and also to simplify
radical expressions. For some radical
expressions, it is possible to do both: First
find a product and then simplify.
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Slide 7- 16
Example Multiply and simplify.
a)
10 6
3 23 4 7
3
b) 9 x y 9 x y
Solution
a)
10 6  60
 4 15
Multiplying radicands
4 is a perfect square.
 2 15
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Slide 7- 17
Solution continued
3 23 4 7 3
7 9
3
b) 9 x y 9 x y  81x y
 3 27  3  x6  x  y9
3 6 3 9 3
3
 27  x  y  3x
 3x 2 y3 3 3x
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Slide 7- 18