An algebraic fraction is a fraction that contains at least one algebraic expression (with a variable) such as $3x$.

The expression can be in the numerator or the denominator or both.

An algebraic fraction is a fraction that contains at least one algebraic expression (with a variable) such as $3x$.

The expression can be in the numerator or the denominator or both.

We can simplify algebraic fractions only when there is a multiplication operation between the algebraic factors in the numerator and the denominator, and there are no addition or subtraction operations.**Steps to simplify algebraic fractions:**

- The first step –

Attempt to factor out a common factor. - The second step –

Attempt to simplify using special product formulas. - The third step –

Attempt to factor by using a trinomial.

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**How do you reduce algebraic fractions?**

- We will find the common factor that is most beneficial for us to extract.
- If we do not find one, we will proceed to factorization using the formulas for shortened multiplication.
- If we cannot use the formulas for shortened multiplication, we will proceed to factorization using trinomials.
- We will simplify (only when there is multiplication between the terms unless the terms are in parentheses, in which case we will treat it as a single term).

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We will make all the denominators the same – we will reach a common denominator.

We will use factorization according to the methods we have learned.

Steps of the operation:

- We will factor all the denominators.
- We will multiply each numerator by the number it needs so that its denominator reaches the common denominator.
- We will write the exercise with one denominator - the common denominator, and between the expressions in the numerators, we will keep the arithmetic operations as in the original exercise.
- After opening parentheses, we might encounter another expression that we need to factor. We will factor it and see if we can simplify.
- We will get a regular fraction and solve it.

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Steps to multiply algebraic fractions:

- Let's try to factor out a common factor.

The common factor can be our variable or any constant number. - If factoring out a common factor is not enough, we will reduce using the formulas for the product of sums or using trinomials.
- Let's find the domain of substitution:

We will set all the denominators we have to 0 and find the solutions.

The domain of substitution will be: x different from what makes the denominator zero. - Let's simplify the fractions.
- We will multiply numerator by numerator and denominator by denominator as in a regular fraction.

Steps for dividing algebraic fractions:

- We will turn the division exercise into a multiplication exercise in this way:

We will leave the first fraction as it is, change the division sign to a multiplication sign, and invert the fraction that comes after the division operation. That is, numerator in place of denominator and denominator in place of numerator. - We will follow the rules for multiplying algebraic fractions:
- We will try to factor out a common factor.

The common factor can be our variable or any free number. - If factoring out a common factor is not enough, we will decompose using the formulas for shortened multiplication and also using trinomials.
- We will find the domain of substitution:

We will set all the denominators we have to 0 and find the solutions.

The domain of substitution will be x different from what zeros the denominator. - We will simplify the fractions.
- We will multiply numerator by numerator and denominator by denominator as in a regular fraction.

- We will try to factor out a common factor.

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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:

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

Determine if the simplification below is correct:

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

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

Determine if the simplification shown below is correct:

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

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

Determine if the simplification below is correct:

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

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

Determine if the simplification below is correct:

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

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

Determine if the simplification below is correct:

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

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

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:

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

Complete the corresponding expression for the denominator

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

We use the formula:

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

We convert the 8 into a fraction, and multiply

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

$16ab\times1=8a$

$16ab=8a$

We divide both sides by 8a:

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

$2b$

$2b$

Determine if the simplification described below is correct:

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

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

Determine if the simplification below is correct:

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

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

Incorrect

Determine if the simplification below is correct:

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

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

Select the field of application of the following fraction:

$\frac{x}{16}$

$All~X$

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:

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

Select the field of application of the following fraction:

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

All numbers

Select the field of application of the following fraction:

$\frac{6}{x}$

All numbers except 0

Select the field of application of the following fraction:

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

$x\neq-2$

Select the field of application of the following fraction:

$\frac{8}{-2+x}$

$x\neq2$

Select the field of application of the following fraction:

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

$x\neq-13$