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Complex Fractions and Simplifying: Lesson 9 (Week 16), Slides of Pre-Calculus

University of San Francisco (USF)Pre-Calculus

A lesson on complex fractions and their simplification. It includes steps for simplifying complex fractions, examples of complex fraction simplification, and important concepts such as the negative exponent rule.

Typology: Slides

2021/2022

Uploaded on 09/27/2022

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16-week Lesson 9 (8-week Lesson 7) Complex Fractions and Simplifying
1
Complex fraction:
- a fraction which has rational expressions in the numerator and/or
denominator
o 1
𝑥 − 1
2
𝑥2− 4 ○ 𝑥
𝑦2 + 𝑦
𝑥2
1
𝑦2 − 1
𝑥2
Steps for Simplifying Complex Fractions
1. simplify the numerator and/or the denominator by adding and/or
subtracting the rational expressions
2. use the procedure for dividing fractions to change division to
multiplication
3. factor the numerator and denominator completely
4. simplify the fraction completely by canceling common factors
𝑥
𝑦2+ 𝑦
𝑥2
1
𝑦2 − 1
𝑥2
𝑥3+𝑦3
𝑥2𝑦2
𝑥2−𝑦2
𝑥2𝑦2
𝑥3+𝑦3
𝑥2𝑦2÷𝑥2−𝑦2
𝑥2𝑦2
𝑥3+𝑦3
𝑥2𝑦2∙𝑥2𝑦2
𝑥2−𝑦2
(𝑥+𝑦)(𝑥2−𝑥𝑦+𝑦2)(𝑥2𝑦2)
(𝑥2𝑦2)(𝑥−𝑦)(𝑥+𝑦)
𝒙𝟐−𝒙𝒚+𝒚𝟐
𝒙−𝒚
Simplifying complex fractions is basically just a
combination of the concepts from the previous three
lessons.
The rational expressions in the numerator and/or
denominator of the complex fraction need to be added
or subtracted first (Lesson 8).
Then the complex fraction gets converted to two
rational expressions being divided, which we don’t
actually divide at all, we simply convert to
multiplication by taking the reciprocal of the divisor
(Lesson 8).
When two fractions are being multiplied, we write
numerator times numerator and denominator times
denominator, but we don’t actually multiply them
because we need to cancel common factors (Lesson
7). Once we factor completely, we can simplify the
rational expressions by canceling common factors
(Lesson 7). Remember that the numerator and
denominator of the rational expression need to be
factored completely in order to simplify, and
factoring we covered in Lessons 6 and 7.
The ability to synthesize all the information from the
previous three lessons is imperative to being able to
simplify complex fractions.
pf3
pf4
pf5
pf8

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Download Complex Fractions and Simplifying: Lesson 9 (Week 16) and more Slides Pre-Calculus in PDF only on Docsity!

Complex fraction:

- a fraction which has rational expressions in the numerator and/or

denominator

o

Steps for Simplifying Complex Fractions

1. simplify the numerator and/or the denominator by adding and/or

subtracting the rational expressions

2. use the procedure for dividing fractions to change division to

multiplication

3. factor the numerator and denominator completely

4. simplify the fraction completely by canceling common factors

÷

Simplifying complex fractions is basically just a

combination of the concepts from the previous three

lessons.

The rational expressions in the numerator and/or

denominator of the complex fraction need to be added

or subtracted first (Lesson 8).

Then the complex fraction gets converted to two

rational expressions being divided, which we don’t

actually divide at all, we simply convert to

multiplication by taking the reciprocal of the divisor

(Lesson 8).

When two fractions are being multiplied, we write

numerator times numerator and denominator times

denominator, but we don’t actually multiply them

because we need to cancel common factors (Lesson

7). Once we factor completely, we can simplify the

rational expressions by canceling common factors

(Lesson 7). Remember that the numerator and

denominator of the rational expression need to be

factored completely in order to simplify, and

factoring we covered in Lessons 6 and 7.

The ability to synthesize all the information from the

previous three lessons is imperative to being able to

simplify complex fractions.

It is imperative that you understand how to simplify, multiply, divide,

add, and subtract rational expressions, as well as how to factor

polynomials, in order to simplify complex fractions. All of those

concepts are combined into one problem when simplifying complex

fractions, and the inability to perform any of those tasks will prevent

you from correctly answering these types of problems.

Example 1 : Perform the indicated operations and simplify the expressions

completely.

a.

1

𝑥

−

1

2

𝑥

2

− 4

b.

𝑥

𝑦

2

−

𝑦

𝑥

2

1

𝑦

2

−

1

𝑥

2

b.

e.

1

𝑥 + ℎ

−

1

𝑥

ℎ

f.

f. 𝑎

𝑥

2

𝑥

2

1

(𝑥+ℎ)

2

1

𝑥

2

(𝑥+ℎ)

2

(𝑥+ℎ)

2

𝑥

2

𝑥

2

(𝑥+ℎ)

2

(𝑥+ℎ)

2

𝑥

2

(𝑥+ℎ)

2

𝑥

2

− (𝑥+ℎ)

2

𝑥

2

(𝑥+ℎ)

2

𝑥

2

− (𝑥+ℎ)(𝑥+ℎ)

𝑥

2

(𝑥+ℎ)

2

𝑥

2

− (𝑥

2

  • 2 𝑥ℎ+ℎ

2

)

𝑥

2

(𝑥+ℎ)

2

𝑥

2

−𝑥

2

− 2 𝑥ℎ−ℎ

2

𝑥

2

( 𝑥+ℎ

)

2

− 2 𝑥ℎ−ℎ

2

𝑥

2

(𝑥+ℎ)

2

2

2

2

÷ ℎ

2

2

2

∙

2

2

2

2

−𝟐𝒙−𝒉

𝒙

( 𝒙+𝒉

)

Negative Exponent Rule:

- once again, to change the sign of an exponent , take the reciprocal of

the factor or expression that has the negative exponent

o 𝑥

3

3

2

Example 2 : Perform the indicated operations and simplify the following

expression completely. Do not include negative exponents in your

answer.

) ÷ (

c.

− 2

− 2

2

d.

− 1

− 2

− 2

− 2

d. a

1

𝑥

2

1

( 𝑥− 1

)

2

1

(𝑥− 1 )

2

(𝑥− 1 )

2

(𝑥− 1 )

2

1

𝑥

2

1

(𝑥− 1 )

2

𝑥

2

𝑥

2

1

( 𝑥− 1

)

2

(𝑥− 1 )

2

−𝑥

2

𝑥

2

( 𝑥− 1

)

2

1

(𝑥− 1 )

2

(𝑥− 1 )(𝑥− 1 )−𝑥

2

𝑥

2

(𝑥− 1 )

2

1

(𝑥− 1 )

2

𝑥

2

− 2 𝑥+ 1 −𝑥

2

𝑥

2

(𝑥− 1 )

2

1

( 𝑥− 1

)

2

1 − 2 𝑥

𝑥

2

(𝑥− 1 )

2

1

(𝑥− 1 )

2

2

2

÷

2

2

2

∙

2

2

2

2

𝟏−𝟐𝒙

𝒙

Answers to Examples:

1a. −

; 1b.

2

2

; 1 c. −

; 1 d.

2

2

2

;

1 e. −

; 1 f. −

2

2

; 2a.

; 2b. −

; 2 c.

2

;

2 d.

;