Transcript Six Sigma 8.8 Mixed Expressions and Complex Fractions CORD Math Mrs. Spitz Fall 2006 Six Sigma Objective • Simplify mixed expressions and complex fractions.

Six Sigma
8.8 Mixed Expressions and
Complex Fractions
CORD Math
Mrs. Spitz
Fall 2006
Six Sigma
Objective
• Simplify mixed expressions and
complex fractions.
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Upcoming
•
•
•
•
•
8.8 Friday 10/27
8.9 Monday 10/30
8.10 Tuesday/Wed
Chapter 8 Review Wed/Thur
Chapter 8 Test Friday
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Assignment
• Pgs. 336-337 #3-29 all
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Introduction
• Algebraic expressions such as
b
a
c
x y
5
x3
are called mixed expressions. Changing
mixed expressions to rational
expression is similar to changing mixed
numbers to improper fractions.
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Mixed number to Improper
Fraction
2
2 3(5) 2
3 or 3  

5
5
5
5
3(5)  2

5
15  2

5
17

5
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Mixed expression to Rational
Expression
a  b a ( a  b) a  b
a


a b
( a  b)
a b
2
2
a(a  b)  a 2  b

(a  b)
a 2  ab  a 2  b

(a  b)
2a 2  ab  b

a b
x y
.
Ex. 1: Find 8  2
2
x y
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2
x 2  y 2 8( x 2  y 2 ) x 2  y 2
8 2
 2
 2
2
2
x y
x y
x  y2
8( x 2  y 2 )  x 2  y 2

2
2
x y
8x  8 y  x  y

x2  y 2
2
2
9x  7 y
 2 2
x y
2
2
2
← Multiply by x2 + y2
representation of 1.
← Combine both 1st and 2nd terms over
the common denominator x2 + y2.
2
← Distribute the 8 using distributive
property.
2
← Combine like terms and simplify.
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What if it has more than one
fraction?
• If a fraction has one or more fractions in
the numerator or denominator, it is
called a complex fraction. Some
complex fractions are shown below:
1
3
2
2
5
3
8
a
b
ab
a
a b
b
1 1

x y
1 1

x y
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Consider the complex fraction.
3
To simplify this fraction, rewrite it as:
5
3
7
7

and proceed as follows:
5 8
8
3 7 3 8 24
   
5 8 5 7 35
Recall that to find the quotient, you multiply by 8/7, the
reciprocal of 7/8.
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Consider the complex fraction.
a
b
c
d
To simplify this fraction, rewrite it as:
a c

b d
3 7

5 8
and proceed as follows:
a c a d ad
   
b d b c bc
Recall that to find the quotient, you multiply by d/c, the
reciprocal of c/d.
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Simplifying Complex Fractions
Rule
• Any complex fraction
a
b
c
d
Where b ≠ 0, c ≠ 0, and
d ≠ 0, may be expressed
as:
ad
bc
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Ex. 2: Simplify
1 1
y
x


x y xy xy

1 1
y
x


x y xy xy
yx
xy

yx
xy
1 1

x y
.
1 1

x y
← The LCD is xy for both the numerator and
the denominator.
← Add to simplify the numerator and subtract
to simplify the denominator.
y  x xy


xy y  x
← Multiply the numerator by the reciprocal of
the denominator.
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Ex. 2: Simplify
y  x xy


xy y  x
yx

yx
1 1

x y
.
1 1

x y
← Eliminate common factors.
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Ex. 3: Simplify
( x  4)( x  4)  1
x4
( x  11)( x  3)  48
x 3
1
x4
x4
48
x  11 
x 3
← The LCD of the numerator is
x + 4, and the LCD of the
denominator is x – 3.
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Ex. 3: Simplify
x  8 x  16  1
x4
2
x  8 x  33  48
x 3
1
x4
x4
48
x  11 
x 3
2
← FOIL the top and don’t forget
to subtract the 1 and add the 48
on the bottom.
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Ex. 3: Simplify
x  8 x  15
x4
2
x  8 x  15
x3
1
x4
x4
48
x  11 
x 3
2
← Simplify by subtracting the 1
in the numerator and adding the
48 in the denominator.
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Ex. 3: Simplify
1
x4
x4
48
x  11 
x 3
x  8 x  15
x 3
 2
x4
x  8 x  15
2
← Multiply by the reciprocal.
x2 + 8x +15 is a common factor
that can be eliminated.
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Ex. 3: Simplify
x3
x4
← Simplify
1
x4
x4
48
x  11 
x 3
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Ex. 4: Simplify
x( x  1)  ( x  4)
x 1
x2
x4
x
x 1
x2
← The LCD of the numerator is
x + 1, and the LCD of the
denominator is x – 2.
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Ex. 4: Simplify
x  xx4
x 1
x2
x4
x
x 1
x2
2
← Distribute and subtract to
simplify the numerator.
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Ex. 4: Simplify
x2  4
x 1
x2
1
x4
x
x 1
x2
← Simplify
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Ex. 4: Simplify
x 4
1


x 1 x  2
x4
x
x 1
x2
2
( x  2)( x  2)

( x  1)( x  2)
( x  2)

( x  1)
← Multiply by the reciprocal.
← x – 2 is the common factor
which can be eliminated.
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