Multiplication and Division of Algebraic Fractions - Examples, Exercises and Solutions

Multiplication and Division Operations in Algebraic Fractions

When we want to multiply or divide algebraic fractions, we will use the same tools that we use for the multiplication or division of common fractions with some small differences.

Steps to carry out for the multiplication of algebraic fractions 1 1 :

  • Let's try to extract the common factor.
    This can be the variable or any free number.
  • If this is not enough, we will factorize with short multiplication formulas or with trinomials.
  • Let's find the solution set.
    • How is the solution set found?
      We will make all the denominators we have equal to 0 0 and find the solution.
      The solution set will be X X : different from what causes our denominator to equal zero.
  • Let's simplify the fractions with determination.
  • Multiply numerator by numerator and denominator by denominator as in any fraction.
Detailed explanation

Suggested Topics to Practice in Advance

  1. Factoring using contracted multiplication
  2. Factorization
  3. Extracting the common factor in parentheses
  4. Factorization: Common factor extraction
  5. Factoring Trinomials

Practice Multiplication and Division of Algebraic Fractions

Question 1

Determine if the simplification below is correct:

\( \frac{5\cdot8}{8\cdot3}=\frac{5}{3} \)

Question 2

Determine if the simplification shown below is correct:

\( \frac{7}{7\cdot8}=8 \)

Question 3

Determine if the simplification below is correct:

\( \frac{4\cdot8}{4}=\frac{1}{8} \)

Question 4

Determine if the simplification below is correct:

\( \frac{3\cdot7}{7\cdot3}=0 \)

Question 5

Determine if the simplification below is correct:

\( \frac{6\cdot3}{6\cdot3}=1 \)

examples with solutions for multiplication and division of algebraic fractions

Exercise #1

Determine if the simplification below is correct:

5⋅88⋅3=53 \frac{5\cdot8}{8\cdot3}=\frac{5}{3}

Video Solution

Step-by-Step Solution

Let's consider the fraction and break it down into two multiplication exercises:

88×53 \frac{8}{8}\times\frac{5}{3}

We simplify:

1×53=53 1\times\frac{5}{3}=\frac{5}{3}

Answer

Correct

Exercise #2

Determine if the simplification shown below is correct:

77⋅8=8 \frac{7}{7\cdot8}=8

Video Solution

Step-by-Step Solution

Let's consider the fraction and break it down into two multiplication exercises:

77×18 \frac{7}{7}\times\frac{1}{8}

We simplify:

1×18=18 1\times\frac{1}{8}=\frac{1}{8}

Therefore, the described simplification is false.

Answer

Incorrect

Exercise #3

Determine if the simplification below is correct:

4⋅84=18 \frac{4\cdot8}{4}=\frac{1}{8}

Video Solution

Step-by-Step Solution

We will divide the fraction exercise into two multiplication exercises:

44×81= \frac{4}{4}\times\frac{8}{1}=

We simplify:

1×81=8 1\times\frac{8}{1}=8

Therefore, the described simplification is false.

Answer

Incorrect

Exercise #4

Determine if the simplification below is correct:

3⋅77⋅3=0 \frac{3\cdot7}{7\cdot3}=0

Video Solution

Step-by-Step Solution

We will divide the fraction exercise into two different multiplication exercises.

As this is a multiplication exercise, you can use the substitution property:

77×33=1×1=1 \frac{7}{7}\times\frac{3}{3}=1\times1=1

Therefore, the simplification described is false.

Answer

Incorrect

Exercise #5

Determine if the simplification below is correct:

6⋅36⋅3=1 \frac{6\cdot3}{6\cdot3}=1

Video Solution

Step-by-Step Solution

We simplify the expression on the left side of the approximate equality:

6̸⋅3̸6̸⋅3̸=?1↓1=!1 \frac{\textcolor{red}{\not{6}}\cdot\textcolor{blue}{\not{3}}}{\textcolor{red}{\not{6}}\cdot\textcolor{blue}{\not{3}}}\stackrel{?}{= }1\\ \downarrow\\ 1\stackrel{!}{= }1 therefore, the described simplification is correct.

Therefore, the correct answer is A.

Answer

Correct

Question 1

Complete the corresponding expression for the denominator

\( \frac{16ab}{?}=8a \)

Question 2

Determine if the simplification described below is correct:

\( \frac{x+6}{y+6}=\frac{x}{y} \)

Question 3

Determine if the simplification below is correct:

\( \frac{3-x}{-x+3}=0 \)

Question 4

Determine if the simplification below is correct:

\( \frac{3\cdot4}{8\cdot3}=\frac{1}{2} \)

Question 5

Select the field of application of the following fraction:

\( \frac{x}{16} \)

examples with solutions for multiplication and division of algebraic fractions

Exercise #1

Complete the corresponding expression for the denominator

16ab?=8a \frac{16ab}{?}=8a

Video Solution

Step-by-Step Solution

We use the formula:

xy=zw→x⋅y=z⋅y \frac{x}{y}=\frac{z}{w}\xrightarrow{}x\cdot y=z\cdot y

We convert the 8 into a fraction, and multiply

16ab?=81 \frac{16ab}{?}=\frac{8}{1}

16ab×1=8a 16ab\times1=8a

16ab=8a 16ab=8a

We divide both sides by 8a:

16ab8a=8a8a \frac{16ab}{8a}=\frac{8a}{8a}

2b 2b

Answer

2b 2b

Exercise #2

Determine if the simplification described below is correct:

x+6y+6=xy \frac{x+6}{y+6}=\frac{x}{y}

Video Solution

Step-by-Step Solution

We use the formula:

x+zy+z=x+zy+z \frac{x+z}{y+z}=\frac{x+z}{y+z}

x+6y+6=x+6y+6 \frac{x+6}{y+6}=\frac{x+6}{y+6}

Therefore, the simplification described is incorrect.

Answer

Incorrect

Exercise #3

Determine if the simplification below is correct:

3−x−x+3=0 \frac{3-x}{-x+3}=0

Video Solution

Step-by-Step Solution

z−x−x+z=1 \frac{z-x}{-x+z}=1

Answer

Incorrect

Exercise #4

Determine if the simplification below is correct:

3⋅48⋅3=12 \frac{3\cdot4}{8\cdot3}=\frac{1}{2}

Video Solution

Step-by-Step Solution

We simplify the expression on the left side of the approximate equality.

First let's consider the fact that the number 8 is a multiple of the number 4:

8=2⋅4 8=2\cdot4
Therefore, we will return to the problem in question and present the number 8 as a multiple of the number 4, then we will simplify the fraction:

3⋅48‾⋅3=?12↓3⋅42⋅4‾⋅3=?12↓3̸⋅4̸2⋅4̸⋅3̸=?12↓12=!12 \frac{3\cdot4}{\underline{8}\cdot3}\stackrel{?}{= }\frac{1}{2}\\ \downarrow\\ \frac{3\cdot4}{\underline{2\cdot4}\cdot3}\stackrel{?}{= }\frac{1}{2}\\ \downarrow\\ \frac{\textcolor{blue}{\not{3}}\cdot\textcolor{red}{\not{4}}}{2\cdot\textcolor{red}{\not{4}}\cdot\textcolor{blue}{\not{3}}}\stackrel{?}{= }\frac{1}{2} \\ \downarrow\\ \frac{1}{2}\stackrel{!}{= }\frac{1}{2}
Therefore, the described simplification is correct.

That is, the correct answer is A.

Answer

True

Exercise #5

Select the field of application of the following fraction:

x16 \frac{x}{16}

Video Solution

Answer

All X All~X

Question 1

Select the field of application of the following fraction:

\( \frac{8+x}{5} \)

Question 2

Select the field of application of the following fraction:

\( \frac{6}{x} \)

Question 3

Select the field of application of the following fraction:

\( \frac{3}{x+2} \)

Question 4

Select the field of application of the following fraction:

\( \frac{8}{-2+x} \)

Question 5

Select the field of application of the following fraction:

\( \frac{7}{13+x} \)

examples with solutions for multiplication and division of algebraic fractions

Exercise #1

Select the field of application of the following fraction:

8+x5 \frac{8+x}{5}

Video Solution

Answer

All numbers

Exercise #2

Select the field of application of the following fraction:

6x \frac{6}{x}

Video Solution

Answer

All numbers except 0

Exercise #3

Select the field of application of the following fraction:

3x+2 \frac{3}{x+2}

Video Solution

Answer

x≠−2 x\neq-2

Exercise #4

Select the field of application of the following fraction:

8−2+x \frac{8}{-2+x}

Video Solution

Answer

x≠2 x\neq2

Exercise #5

Select the field of application of the following fraction:

713+x \frac{7}{13+x}

Video Solution

Answer

x≠−13 x\neq-13

Topics learned in later sections

  1. Algebraic Fractions
  2. Simplifying Algebraic Fractions
  3. Factoring Algebraic Fractions
  4. Addition and Subtraction of Algebraic Fractions
  5. Solving Equations by Factoring
  6. Uses of Factorization