Select the domain of the following fraction: $$ \frac{8+x}{5} $$

All decomposed numbers except -5

Multiplication and Division of Algebraic Fractions

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.

Detailed explanation

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Test yourself on factorization and algebraic fractions!

Complete the corresponding expression for the denominator

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

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Steps to carry out for the division of algebraic fractions $2$ :

We will convert the division exercise into a multiplication one, as we do with common fractions.
How will we do it correctly?
We will leave the first fraction as it is, change the division sign to a multiplication sign, and invert the fraction that appears after the sign. That is, numerator instead of denominator and denominator instead of numerator.
We will act according to the rules of multiplication of algebraic fractions:
- Let's try to extract the common factor.
  This can be the unknown or any free number.
- If this is not enough, we will factorize using short multiplication formulas and 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.

Let's look at an example of multiplying algebraic fractions

$\frac{x+2}{x+3}\times \frac{3x+9}{x^2-4}=$

Let's try to factorize by extracting the common factor and with the shortcut multiplication formulas, and we will obtain:
$\frac{x+2}{x+3}\times \frac{3(x+3)}{(x-2)(x+2}=$

Let's find the solution set:

$x≠-3,2,-2$

Let's reduce the fractions and we will obtain:

Let's see an example of multiplying algebraic fractions

$1\times \frac{3}{(x-2)}=$
Multiply and it will give us:
$\frac{3}{x-2}$

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Test your knowledge

Question 1

Complete the corresponding expression for the denominator

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

Question 2

Complete the corresponding expression for the denominator

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

Question 3

Complete the corresponding expression for the denominator

$\frac{19ab}{?}=a$

Example of Division of Algebraic Fractions

$\frac{x^2-8x+15}{x^2-3x+2}:\frac{x^2-9}{x-1}=$

Let's convert the division exercise into a multiplication one:

$\frac{x^2-8x+15}{x^2-3x+2}\times \frac{x-1}{x^{2-9}}=$
Now, let's factor and we will get:
$\frac{(x-5)(x-3)}{(x-2)(x-1)}\times \frac{x-1}{(x-3)(x+3)}=$
Let's find the solution set:
$x≠2,1,3,-3$

Let's simplify, we will get:

$\frac{x-5}{x-2}\times \frac{1}{x+3}$

Let's multiply and it will give us:
$\frac{x-5}{(x-2)(x+3)}$

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Examples and exercises with solutions for multiplication and division of algebraic fractions

Exercise #1

Complete the corresponding expression for the denominator

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

Video Solution

Step-by-Step Solution

Let's examine the problem:

$\frac{12ab}{?}=1$ Now let's think logically, and remember the known fact that dividing any number by itself always yields the result 1,

Therefore, in order to get the result 1 from dividing two numbers, the only way is to divide the number by itself, meaning-

The missing expression in the denominator of the fraction on the left side is the complete expression that appears in the numerator of the same fraction:

$12ab$ .

Therefore- the correct answer is answer D.

Answer

$12ab$

Exercise #2

Complete the corresponding expression for the denominator

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

Video Solution

Step-by-Step Solution

After examining the problem, proceed to write the expression on the right side as a fraction (using the fact that dividing a number by 1 does not change its value):

$\frac{16ab}{?}=2b \\ \downarrow\\ \frac{16ab}{?}=\frac{2b}{1}$

Remember the fraction reduction operation,

In order for the fraction on the left side to be deemed reducible, we want all the terms in its denominator to have a common factor. Additionally, we want to reduce the number 16 in order to obtain the number 2. Furthermore we want to reduce the term $a$ from the fraction's denominator given that in the expression on the right side it does not appear. Therefore we will choose the expression:

$8a$

Due to the fact that:

$16=8\cdot 2$

Let's verify that with this choice we indeed obtain the expression on the right side:

$\frac{16ab}{?}=\frac{2b}{1} \\ \downarrow\\ \frac{\not{16}\not{a}b}{\textcolor{red}{\not{8}\not{a}}}\stackrel{?}{= }\frac{2b}{1} \\ \downarrow\\ \boxed{\frac{2b}{1}\stackrel{!}{= }\frac{2b}{1} }$

Therefore this choice is indeed correct.

In other words - the correct answer is answer B.

Answer

$8a$

Exercise #3

Complete the corresponding expression for the denominator

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

Video Solution

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 #4

Complete the corresponding expression for the denominator

$\frac{19ab}{?}=a$

Video Solution

Step-by-Step Solution

Upon examining the problem, proceed to write down the expression on the right side as a fraction (using the fact that dividing a number by 1 doesn't change its value):

$\frac{19ab}{?}=a \\ \downarrow\\ \frac{19ab}{?}=\frac{a}{1}$
Remember the fraction reduction operation,

In order for the fraction on the left side to be deemed reducible, we want all the terms in its denominator to have a common factor. Additionally, we want to reduce the number 19 in order to obtain the number 1 as well as reducing the term $b$ from the fraction's numerator given that in the expression on the right side it doesn't appear. Therefore we'll choose the expression:

$19b$

Let's verify that this choice results in the expression on the right side:

$\frac{19ab}{?}=\frac{a}{1} \\ \downarrow\\ \frac{\not{19}a\not{b}}{\textcolor{red}{\not{19}\not{b}}}\stackrel{?}{= }\frac{a}{1} \\ \downarrow\\ \boxed{\frac{a}{1}\stackrel{!}{= }\frac{a}{1} }$

Therefore this choice is indeed correct.

In other words - the correct answer is answer D.

Answer

$19b$

Exercise #5

Complete the corresponding expression for the denominator

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

Video Solution

Step-by-Step Solution

Upon examining the problem, proceed to write the expression on the right side as a fraction (using the fact that dividing a number by 1 does not change its value):

$\frac{27ab}{\text{?}}=3ab\\ \downarrow\\ \frac{27ab}{\text{?}}=\frac{3ab}{1}$

Remember the fraction reduction operation,

Note that both in the numerator of the expression on the right side and in the numerator of the expression on the left side the expression $ab$ is present. Therefore in the expression we are looking for there are no variables (since we are not interested in reducing them from the expression in the numerator on the left side),

Next, determine which number was chosen to be in the denominator of the expression on the left side in order that its reduction with the number 27 yields the number 3. The answer to this - the number 9,

Due to the fact that:

$27=9\cdot 3$

Let's verify that this choice indeed gives us the expression on the right side:

$\frac{27ab}{\text{?}}=\frac{3ab}{1} \\ \downarrow\\ \frac{\not{27}ab}{\textcolor{red}{\not{9}}}\stackrel{?}{= }\frac{3ab}{1} \\ \downarrow\\ \boxed{\frac{3ab}{1}\stackrel{!}{= }\frac{3ab}{1} }$

Therefore this choice is indeed correct.

In other words - the correct answer is answer A.

Answer

$9$

Do you know what the answer is?

Question 1

Complete the corresponding expression for the denominator

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

Question 2

Determine if the simplification below is correct:

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

Question 3

Determine if the simplification below is correct:

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

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Multiplication and Division of Algebraic Fractions

Multiplication and Division Operations in Algebraic Fractions

Test yourself on factorization and algebraic fractions!

Let's look at an example of multiplying algebraic fractions

Example of Division of Algebraic Fractions

Examples and exercises with solutions for multiplication and division of algebraic fractions

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

More Questions

Factorization and Algebraic Fractions