16 the multiplier method for simplifying complex fractions

$: 16 the multiplier method for simplifying complex fractions$
Complex Fractions

Frank Ma © 2011

Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction.

Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.

Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.We want to simplify complex fractions to regular fractions.

Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.We want to simplify complex fractions to regular fractions.The “easy” complex fractions are just regular divisions of of a fraction by another fraction.

Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.We want to simplify complex fractions to regular fractions.The “easy” complex fractions are just regular divisions of of a fraction by another fraction. In this case just flip the denominator then simplify the multiplication,

Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.We want to simplify complex fractions to regular fractions.The “easy” complex fractions are just regular divisions of of a fraction by another fraction. In this case just flip the denominator then simplify the multiplication, i.e.

ABCD


ABCD

AB C

D*

flip


ABCD

AB C

D*

flip

Example A. Simplify

4x2y9xy2

6


ABCD

AB C

D*

flip

Example A. Simplify

4x2y9xy2

6

=

flip

4x2y9 xy2

6


ABCD

AB C

D*

flip

Example A. Simplify

4x2y9xy2

6

=

flip

4x2y9 xy2

63

2


ABCD

AB C

D*

flip

Example A. Simplify

4x2y9xy2

6

=

flip

4x2y9 xy2

63

2x


ABCD

AB C

D*

flip

Example A. Simplify

4x2y9xy2

6

=

flip

4x2y9 xy2

63

2x

y


ABCD

AB C

D*

flip

Example A. Simplify

4x2y9xy2

6

=

flip

4x2y9 xy2

6 =3

2x

y

8x3y

Complex FractionsWe give two methods for simplifying general complex fractions.

Complex FractionsWe give two methods for simplifying general complex fractions. The first method is to reduce the problem to an “easy” problem.

Complex FractionsWe give two methods for simplifying general complex fractions. The first method is to reduce the problem to an “easy” problem. We do this by combining the numerator into one fraction and the denominator into one fraction.

Complex FractionsWe give two methods for simplifying general complex fractions. The first method is to reduce the problem to an “easy” problem. We do this by combining the numerator into one fraction and the denominator into one fraction. We can use cross multiplication for combining two fractions.

Example B. Simplify

Complex FractionsWe give two methods for simplifying general complex fractions. The first method is to reduce the problem to an “easy” problem. We do this by combining the numerator into one fraction and the denominator into one fraction. We can use cross multiplication for combining two fractions. x

x + 1 + 1

1 – xx – 1

Example B. Simplify


x + 1 + 1

1 – xx – 1 x

x + 1 + 1

1 – xx – 1

Example B. Simplify


x + 1 + 1

1 – xx – 1 x

x + 1 + 1

1 – xx – 1

=

xx + 1 +

– xx – 1

11

11

Example B. Simplify


x + 1 + 1

1 – xx – 1 x

x + 1 + 1

1 – xx – 1

=

xx + 1 +

– xx – 1

11

11

=

x + (x + 1)x + 1

Example B. Simplify


x + 1 + 1

1 – xx – 1 x

x + 1 + 1

1 – xx – 1

=

xx + 1 +

– xx – 1

11

11

=

x + (x + 1)x + 1(x – 1) – x

x – 1

Example B. Simplify


x + 1 + 1

1 – xx – 1 x

x + 1 + 1

1 – xx – 1

=

xx + 1 +

– xx – 1

11

11

=

x + (x + 1)x + 1(x – 1) – x

x – 1

=

2x + 1x + 1

–1x – 1

Complex Fractions

xx + 1 + 1

1 – xx – 1

=

2x + 1x + 1

–1x – 1

Therefore,

Complex Fractions

xx + 1 + 1

1 – xx – 1

=

2x + 1x + 1

–1x – 1

Therefore,

= (2x + 1)(x + 1)

(x – 1) (–1)

Complex Fractions

xx + 1 + 1

1 – xx – 1

=

2x + 1x + 1

–1x – 1

Therefore,

= (2x + 1)(x + 1)

(x – 1) = – (2x + 1)(x + 1)

(x – 1) (–1)

Complex Fractions

xx + 1 + 1

1 – xx – 1

=

Therefore,

The second method is to multiply the LCD of all the terms to the numerator and the denominator of the complex fraction.

2x + 1x + 1

–1x – 1

= (2x + 1)(x + 1)

(x – 1) = – (2x + 1)(x + 1)

(x – 1) (–1)

Complex Fractions

xx + 1 + 1

1 – xx – 1

=

Therefore,


Example C. Simplify

32 + 1

1 –

– 56

23

34+

2x + 1x + 1

–1x – 1

= (2x + 1)(x + 1)

(x – 1) = – (2x + 1)(x + 1)

(x – 1) (–1)

Complex Fractions

xx + 1 + 1

1 – xx – 1

=

Therefore,


Example C. Simplify

32 + 1

1 –

– 56

23

34+

The LCD of 23

32

56

34, , , is 12.

2x + 1x + 1

–1x – 1

= (2x + 1)(x + 1)

(x – 1) = – (2x + 1)(x + 1)

(x – 1) (–1)

Complex Fractions

xx + 1 + 1

1 – xx – 1

=

Therefore,


Example C. Simplify

32 + 1

1 –

– 56

23

34+

The LCD of 23

32

56

34, , , is 12.

Multiply 12 to the top and bottom of the complex fraction.32 + 1

1 –

– 56

23

34+

2x + 1x + 1

–1x – 1

= (2x + 1)(x + 1) (–1)

(x – 1) = – (2x + 1)(x + 1)

(x – 1)

Complex Fractions

xx + 1 + 1

1 – xx – 1

=

Therefore,


Example C. Simplify

32 + 1

1 –

– 56

23

34+

The LCD of 23

32

56

34, , , is 12.


1 –

– 56

23

34+ (

( ))

12

12

2x + 1x + 1

–1x – 1

= (2x + 1)(x + 1) (–1)

(x – 1) = – (2x + 1)(x + 1)

(x – 1)

Complex Fractions

xx + 1 + 1

1 – xx – 1

=

Therefore,


Example C. Simplify

32 + 1

1 –

– 56

23

34+

The LCD of 23

32

56

34, , , is 12.


1 –

– 56

23

34+ (

( ))

12

12

2x + 1x + 1

–1x – 1

= (2x + 1)(x + 1)

(x – 1) = – (2x + 1)(x + 1)

(x – 1)

This is OK since is 1.12121

(–1)

Complex Fractions

xx + 1 + 1

1 – xx – 1

=

Therefore,


Example C. Simplify

32 + 1

1 –

– 56

23

34+

The LCD of 23

32

56

34, , , is 12.


1 –

– 56

23

34+ (

( ))

12

12

6

2x + 1x + 1

–1x – 1

= (2x + 1)(x + 1)

(x – 1) = – (2x + 1)(x + 1)

(x – 1)

Distribute the multiplication.

(–1)

Complex Fractions

xx + 1 + 1

1 – xx – 1

=

Therefore,


Example C. Simplify

32 + 1

1 –

– 56

23

34+

The LCD of 23

32

56

34, , , is 12.


1 –

– 56

23

34+ (

( ))

12

12

6 12

2x + 1x + 1

–1x – 1

= (2x + 1)(x + 1)

(x – 1) = – (2x + 1)(x + 1)

(x – 1)


(–1)

Complex Fractions

xx + 1 + 1

1 – xx – 1

=

Therefore,


Example C. Simplify

32 + 1

1 –

– 56

23

34+

The LCD of 23

32

56

34, , , is 12.


1 –

– 56

23

34+ (

( ))

12

12

6 12 2

2x + 1x + 1

–1x – 1

= (2x + 1)(x + 1)

(x – 1) = – (2x + 1)(x + 1)

(x – 1)


(–1)

Complex Fractions

xx + 1 + 1

1 – xx – 1

=

Therefore,


Example C. Simplify

32 + 1

1 –

– 56

23

34+

The LCD of 23

32

56

34, , , is 12.


1 –

– 56

23

34+ (

( ))

12

12

6 12 2

12 4 3

2x + 1x + 1

–1x – 1

= (2x + 1)(x + 1)

(x – 1) = – (2x + 1)(x + 1)

(x – 1)


(–1)

Complex Fractions

xx + 1 + 1

1 – xx – 1

=

Therefore,


Example C. Simplify

32 + 1

1 –

– 56

23

34+

The LCD of 23

32

56

34, , , is 12.


1 –

– 56

23

34+ (

( ))

12

12=

6 12 2

12 4 318 + 12 – 1012 – 8 + 9

2x + 1x + 1

–1x – 1

= (2x + 1)(x + 1)

(x – 1) = – (2x + 1)(x + 1)

(x – 1) (–1)

Complex Fractions

xx + 1 + 1

1 – xx – 1

=

Therefore,


Example C. Simplify

32 + 1

1 –

– 56

23

34+

The LCD of 23

32

56

34, , , is 12.


1 –

– 56

23

34+ (

( ))

12

12=

6 12 2

12 4 318 + 12 – 1012 – 8 + 9 = 20

13

2x + 1x + 1

–1x – 1

= (2x + 1)(x + 1)

(x – 1) = – (2x + 1)(x + 1)

(x – 1) (–1)

Complex Fractionsx2

+ 1

– 1 Example D. Simplify x

y

y2

Complex Fractionsx2

+ 1

– 1 Example D. Simplify

The LCD of

xy

is y2.

y2

x2

xy y2 ,

Complex Fractionsx2

+ 1


The LCD of

xy

is y2.

Multiply the LCD to top and bottom of the complex fraction.

y2

x2

xy y2 ,

x2

+ 1

– 1

xy

y2

Complex Fractionsx2

+ 1


The LCD of

xy

is y2.


y2

x2

xy y2 ,

x2

+ 1

– 1

xy

y2

((

)) y2

y2

Complex Fractionsx2

+ 1


The LCD of

xy

is y2.


y2

x2

xy y2 ,

x2

+ 1

– 1

xy

y2

((

)) y2

y2

1

Complex Fractionsx2

+ 1


The LCD of

xy

is y2.


y2

x2

xy y2 ,

x2

+ 1

– 1

xy

y2

((

)) y2

y2

y21

Complex Fractionsx2

+ 1


The LCD of

xy

is y2.


y2

x2

xy y2 ,

x2

+ 1

– 1

xy

y2

((

)) y2

y2

y2

y2y

1

Complex Fractionsx2

+ 1


The LCD of

xy

is y2.


y2

x2

xy y2 ,

x2

+ 1

– 1

xy

y2

((

)) y2

y2

=

y2

y2y

1

x2 – y2

xy + y2

Complex Fractionsx2

+ 1


The LCD of

xy

is y2.


y2

x2

xy y2 ,

x2

+ 1

– 1

xy

y2

((

)) y2

y2

=

y2

y2y

1

x2 – y2

xy + y2

= (x – y)(x + y)y(x + y)

To simplify this, put it in the factored form .

Complex Fractionsx2

+ 1


The LCD of

xy

is y2.


y2

x2

xy y2 ,

x2

+ 1

– 1

xy

y2

((

)) y2

y2

=

y2

y2y

1

x2 – y2

xy + y2 To simplify this, put it in the factored form .

= (x – y)(x + y)y(x + y)

Complex Fractionsx2

+ 1


The LCD of

xy

is y2.


y2

x2

xy y2 ,

x2

+ 1

– 1

xy

y2

((

)) y2

y2

=

y2

y2y

1

x2 – y2

xy + y2 To simplify this, put it in the factored form .

= (x – y)(x + y)y(x + y)

= x – y y

Complex Fractionsx

x – 2

–

+

3x + 2

Example E. Simplify

3x + 2

xx – 2

Complex Fractionsx

x – 2

–

+

3x + 2

Example E. Simplify

The LCD of

3x + 2

xx – 2

xx – 2

xx – 2

3x + 2

3x + 2 , , , is (x – 2)(x + 2).

Complex Fractionsx

x – 2

–

+

3x + 2

Example E. Simplify

The LCD of

3x + 2

xx – 2

xx – 2

xx – 2

3x + 2

3x + 2 , , , is (x – 2)(x + 2).


xx – 2

–

+

3x + 2

3x + 2

xx – 2

Complex Fractionsx

x – 2

–

+

3x + 2

Example E. Simplify

The LCD of

3x + 2

xx – 2

xx – 2

xx – 2

3x + 2

3x + 2 , , , is (x – 2)(x + 2).


xx – 2

–

+

3x + 2

3x + 2

xx – 2 (

( ))

(x – 2)(x + 2)

(x – 2)(x + 2)

Complex Fractionsx

x – 2

–

+

3x + 2

Example E. Simplify

The LCD of

3x + 2

xx – 2

xx – 2

xx – 2

3x + 2

3x + 2 , , , is (x – 2)(x + 2).


xx – 2

–

+

3x + 2

3x + 2

xx – 2 (

( ))

(x – 2)(x + 2)

(x – 2)(x + 2)

(x + 2)

Complex Fractionsx

x – 2

–

+

3x + 2

Example E. Simplify

The LCD of

3x + 2

xx – 2

xx – 2

xx – 2

3x + 2

3x + 2 , , , is (x – 2)(x + 2).


xx – 2

–

+

3x + 2

3x + 2

xx – 2 (

( ))

(x – 2)(x + 2)

(x – 2)(x + 2)

(x + 2) (x – 2)

Complex Fractionsx

x – 2

–

+

3x + 2

Example E. Simplify

The LCD of

3x + 2

xx – 2

xx – 2

xx – 2

3x + 2

3x + 2 , , , is (x – 2)(x + 2).


xx – 2

–

+

3x + 2

3x + 2

xx – 2 (

( ))

(x – 2)(x + 2)

(x – 2)(x + 2)

=

(x + 2)

(x + 2)

(x – 2)

(x – 2)

x(x + 2) + 3(x – 2)x(x + 2) – 3(x – 2) Expand and put the fraction

in the factored form.

Complex Fractionsx

x – 2

–

+

3x + 2

Example E. Simplify

The LCD of

3x + 2

xx – 2

xx – 2

xx – 2

3x + 2

3x + 2 , , , is (x – 2)(x + 2).


xx – 2

–

+

3x + 2

3x + 2

xx – 2 (

( ))

(x – 2)(x + 2)

(x – 2)(x + 2)

=

(x + 2)

(x + 2)

(x – 2)

(x – 2)


in the factored form. = x2 + 2x + 3x – 6

x2 + 2x – 3x + 6

Complex Fractionsx

x – 2

–

+

3x + 2

Example E. Simplify

The LCD of

3x + 2

xx – 2

xx – 2

xx – 2

3x + 2

3x + 2 , , , is (x – 2)(x + 2).


xx – 2

–

+

3x + 2

3x + 2

xx – 2 (

( ))

(x – 2)(x + 2)

(x – 2)(x + 2)

=

(x + 2)

(x + 2)

(x – 2)

(x – 2)



x2 + 2x – 3x + 6

= x2 + 5x – 6x2 – x + 6

Complex Fractionsx

x – 2

–

+

3x + 2

Example E. Simplify

The LCD of

3x + 2

xx – 2

xx – 2

xx – 2

3x + 2

3x + 2 , , , is (x – 2)(x + 2).


xx – 2

–

+

3x + 2

3x + 2

xx – 2 (

( ))

(x – 2)(x + 2)

(x – 2)(x + 2)

=

(x + 2)

(x + 2)

(x – 2)

(x – 2)



x2 + 2x – 3x + 6

= x2 + 5x – 6x2 – x + 6

This is the simplified.= (x + 6)(x – 1)x2 – x + 6

Complex Fractions2

x – 1

–

+

3x + 2

Example F. Simplify

3x + 2

1x – 2

Complex Fractions2

x – 1

–

+

3x + 2

Example F. Simplify

3x + 2

1x – 2

Use the crossing method.

Complex Fractions2

x – 1

–

+

3x + 2

Example F. Simplify

3x + 2

1x – 2

Use the crossing method.2

x – 1

–

+

3x + 2

3x + 2

1x – 2

Complex Fractions2

x – 1

–

+

3x + 2

Example F. Simplify

3x + 2

1x – 2


x – 1

–

+

3x + 2

3x + 2

1x – 2

=

2(x + 2) + 3(x – 1)(x – 1)(x + 2)

2(x + 3) – 3(x – 2)(x – 2)(x + 2)

Complex Fractions2

x – 1

–

+

3x + 2

Example F. Simplify

3x + 2

1x – 2


x – 1

–

+

3x + 2

3x + 2

1x – 2

=

2(x + 2) + 3(x – 1)(x – 1)(x + 2)

2(x + 3) – 3(x – 2)(x – 2)(x + 2)

=

5x +1(x – 1)(x + 2) – x + 12

(x – 2)(x + 2)

Complex Fractions2

x – 1

–

+

3x + 2

Example F. Simplify

3x + 2

1x – 2


x – 1

–

+

3x + 2

3x + 2

1x – 2

=

2(x + 2) + 3(x – 1)(x – 1)(x + 2)

2(x + 3) – 3(x – 2)(x – 2)(x + 2)

flip the bottom fraction

=

5x +1(x – 1)(x + 2) – x + 12

(x – 2)(x + 2)

Complex Fractions2

x – 1

–

+

3x + 2

Example F. Simplify

3x + 2

1x – 2


x – 1

–

+

3x + 2

3x + 2

1x – 2

=

2(x + 2) + 3(x – 1)(x – 1)(x + 2)

2(x + 3) – 3(x – 2)(x – 2)(x + 2)


=

5x +1(x – 1)(x + 2) – x + 12

(x – 2)(x + 2)

=(5x +1)

(x – 1)(x + 2) (– x + 12)(x – 2)(x + 2)

Complex Fractions2

x – 1

–

+

3x + 2

Example F. Simplify

3x + 2

1x – 2


x – 1

–

+

3x + 2

3x + 2

1x – 2

=

2(x + 2) + 3(x – 1)(x – 1)(x + 2)

2(x + 3) – 3(x – 2)(x – 2)(x + 2)


=

5x +1(x – 1)(x + 2) – x + 12

(x – 2)(x + 2)

=(5x +1)

(x – 1)(x + 2) (– x + 12)(x – 2)(x + 2)

=(5x + 1)(x – 2)(x – 1)(–x + 12)

Complex FractionsEx. A. Simplify.

1.

2x2y54y2 2.

34x2

3x2 3. 4x2

2x3 4.

12x2

5y4xy15

3

+ 1

–

13

45.

2 3

5

–

–

23

66.

4 314

4

+

–

16

57.

11 15712

1

+ 1

– 2

2y

x8.1

+ 2

1 –

1x2

xy9.1

+

–

1x

y10.1x

1y

2

–

–

4x2

y11.2x

4y2

2

–

–

4x2

y12.2x

4y2

2x + 1 + 1

1 – 1x – 1

13.

Complex Fractions1

2x + 1 – 2

3 – 1x + 2

14.

2x – 3 – 1

2 – 12x + 1

15.

2x + 3 –

+ 1x + 3

16.

3x + 2

3x + 2

–2 2x + 1 –

+ 3x + 4

17.

1x + 4

22x + 1

2 x + 2 –

+ 2x + 5

18.

13x – 1

3x + 2

4 2x + 3 –

+ 3x + 4

19.

33x – 2

53x – 2

–5 2x + 5 –

+ 3–x + 4

20.

22x – 3

62x – 3

23 + 2

2 –

– 16

23

12+

21.

12 – + 56

23

14–

22.34

32 +

Complex Fractions

23.

2x – 1

–

+

3x + 3

xx + 3

xx – 1

24.

3x + 2

–

+

3x + 2

xx – 2

xx – 2

25.2

x + h – 2x

h26.

3x – h– 3

xh

27.2

x + h – 2x – h

h28.

3x + h–

h

3x – h

16 the multiplier method for simplifying complex fractions

Education