# 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 $1$:

• 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 $0$ and find the solution.
The solution set will be $X$: 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.

## Examples with solutions for Multiplication and Division of Algebraic Fractions

### Exercise #1

Determine if the simplification shown below is correct:

$\frac{7}{7\cdot8}=8$

### Step-by-Step Solution

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

$\frac{7}{7}\times\frac{1}{8}$

We simplify:

$1\times\frac{1}{8}=\frac{1}{8}$

Therefore, the described simplification is false.

Incorrect

### Exercise #2

Determine if the simplification below is correct:

$\frac{4\cdot8}{4}=\frac{1}{8}$

### Step-by-Step Solution

We will divide the fraction exercise into two multiplication exercises:

$\frac{4}{4}\times\frac{8}{1}=$

We simplify:

$1\times\frac{8}{1}=8$

Therefore, the described simplification is false.

Incorrect

### Exercise #3

Determine if the simplification below is correct:

$\frac{3\cdot7}{7\cdot3}=0$

### 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:

$\frac{7}{7}\times\frac{3}{3}=1\times1=1$

Therefore, the simplification described is false.

Incorrect

### Exercise #4

Determine if the simplification below is correct:

$\frac{5\cdot8}{8\cdot3}=\frac{5}{3}$

### Step-by-Step Solution

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

$\frac{8}{8}\times\frac{5}{3}$

We simplify:

$1\times\frac{5}{3}=\frac{5}{3}$

Correct

### Exercise #5

Determine if the simplification below is correct:

$\frac{6\cdot3}{6\cdot3}=1$

### Step-by-Step Solution

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

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

Therefore, the correct answer is A.

Correct

### Exercise #6

Select the field of application of the following fraction:

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

### Step-by-Step Solution

Since the domain depends on the denominator, we note that there is no variable in the denominator.

Therefore, the domain is all numbers.

All numbers

### Exercise #7

Select the field of application of the following fraction:

$\frac{6}{x}$

### Step-by-Step Solution

Since the domain of definition depends on the denominator, and X appears in the denominator

All numbers will be suitable except for 0.

In other words, the domain of definition:

$x\ne0$

### Answer

All numbers except 0

### Exercise #8

Complete the corresponding expression for the denominator

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

### Step-by-Step Solution

Using the formula:

$\frac{x}{y}=\frac{z}{w}\xrightarrow{}x\cdot y=z\cdot y$

We first convert the 8 into a fraction, and multiply

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

$16ab\times1=8a$

$16ab=8a$

We then divide both sides by 8a:

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

$2b$

### Answer

$2b$

### Exercise #9

Determine if the simplification described below is correct:

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

### Step-by-Step Solution

We use the formula:

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

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

Therefore, the simplification described is incorrect.

Incorrect

### Exercise #10

Determine if the simplification below is correct:

$\frac{3-x}{-x+3}=0$

### Step-by-Step Solution

$\frac{z-x}{-x+z}=1$

Incorrect

### Exercise #11

Indicate whether true or false

$\frac{c\cdot a}{a\cdot c}=0$

### Step-by-Step Solution

Let's simplify the expression on the left side of the proposed equation:

$\frac{\not{c}\cdot \not{a}}{\not{a}\cdot \not{c}}\stackrel{?}{= }0 \\ 1 \stackrel{?}{= }0$Clearly, we get a false statement because: 1 is different from: 0

$\boxed{ 1 \stackrel{!}{\neq }0}$Therefore, the proposed equation is not correct,

Which means the correct answer is answer B.

Not true

### Exercise #12

Determine if the simplification below is correct:

$\frac{3\cdot4}{8\cdot3}=\frac{1}{2}$

### 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:

$8=2\cdot4$
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:

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

That is, the correct answer is A.

True

### Exercise #13

Complete the corresponding expression for the denominator

$\frac{12ab}{?}=1$

### Answer

$12ab$

### Exercise #14

Complete the corresponding expression for the denominator

$\frac{16ab}{?}=2b$

### Answer

$8a$

### Exercise #15

Complete the corresponding expression for the denominator

$\frac{27ab}{\text{?}}=3ab$

### Answer

$9$