Converting rational expressions. Video lesson “Converting rational expressions How to solve rational expressions

This lesson will cover basic information about rational expressions and their transformations, as well as examples of transformations of rational expressions. This topic summarizes the topics we have studied so far. Transformations of rational expressions involve addition, subtraction, multiplication, division, exponentiation of algebraic fractions, reduction, factorization, etc. As part of the lesson, we will look at what a rational expression is, and also analyze examples of their transformation.

Subject:Algebraic fractions. Arithmetic operations on algebraic fractions

Lesson:Basic information about rational expressions and their transformations

Definition

Rational expression is an expression consisting of numbers, variables, arithmetic operations and the operation of exponentiation.

Let's look at an example of a rational expression:

Special cases of rational expressions:

1st degree: ;

2. monomial: ;

3. fraction: .

Converting a rational expression is a simplification of a rational expression. The order of actions when transforming rational expressions: first there are operations in brackets, then multiplication (division) operations, and then addition (subtraction) operations.

Let's look at several examples of transforming rational expressions.

Example 1

Solution:

Let's solve this example step by step. The action in parentheses is executed first.

Answer:

Example 2

Solution:

Answer:

Example 3

Solution:

Answer: .

Note: Perhaps, when you saw this example, an idea arose: reduce the fraction before reducing it to a common denominator. Indeed, it is absolutely correct: first it is advisable to simplify the expression as much as possible, and then transform it. Let's try to solve the same example in the second way.

As you can see, the answer turned out to be absolutely similar, but the solution turned out to be somewhat simpler.

In this lesson we looked at rational expressions and their transformations, as well as several specific examples of these transformations.

References

1. Bashmakov M.I. Algebra 8th grade. - M.: Education, 2004.

2. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. and others. Algebra 8. - 5th ed. - M.: Education, 2010.

Any fractional expression (clause 48) can be written in the form , where P and Q are rational expressions, and Q necessarily contains variables. Such a fraction is called a rational fraction.

Examples of rational fractions:

The main property of a fraction is expressed by an identity that is fair under the conditions here - a whole rational expression. This means that the numerator and denominator of a rational fraction can be multiplied or divided by the same non-zero number, monomial or polynomial.

For example, the property of a fraction can be used to change the signs of the terms of a fraction. If the numerator and denominator of a fraction are multiplied by -1, we get Thus, the value of the fraction will not change if the signs of the numerator and denominator are simultaneously changed. If you change the sign of only the numerator or only the denominator, then the fraction will change its sign:

For example,

60. Reducing rational fractions.

To reduce a fraction means to divide the numerator and denominator of the fraction by a common factor. The possibility of such a reduction is due to the basic property of the fraction.

In order to reduce a rational fraction, you need to factor the numerator and denominator. If it turns out that the numerator and denominator have common factors, then the fraction can be reduced. If there are no common factors, then converting a fraction through reduction is impossible.

Example. Reduce a fraction

Solution. We have

The reduction of a fraction is carried out under the condition .

61. Reducing rational fractions to a common denominator.

The common denominator of several rational fractions is the whole rational expression, which is divided by the denominator of each fraction (see paragraph 54).

For example, the common denominator of fractions is a polynomial since it is divisible by both and by and polynomial and polynomial and polynomial, etc. Usually they take such a common denominator that any other common denominator is divisible by Echosen. This simplest denominator is sometimes called the lowest common denominator.

In the example discussed above, the common denominator is We have

Reducing these fractions to a common denominator is achieved by multiplying the numerator and denominator of the first fraction by 2. and the numerator and denominator of the second fraction by Polynomials are called additional factors for the first and second fractions, respectively. The additional factor for a given fraction is equal to the quotient of dividing the common denominator by the denominator of the given fraction.

To reduce several rational fractions to a common denominator, you need:

1) factor the denominator of each fraction;

2) create a common denominator by including as factors all the factors obtained in step 1) of the expansions; if a certain factor is present in several expansions, then it is taken with an exponent equal to the largest of the available ones;

3) find additional factors for each of the fractions (for this, the common denominator is divided by the denominator of the fraction);

4) by multiplying the numerator and denominator of each fraction by an additional factor, bring the fraction to a common denominator.

Example. Reduce a fraction to a common denominator

Solution. Let's factorize the denominators:

The following factors must be included in the common denominator: and the least common multiple of the numbers 12, 18, 24, i.e. This means that the common denominator has the form

Additional factors: for the first fraction for the second for the third. So, we get:

62. Addition and subtraction of rational fractions.

The sum of two (and in general any finite number) rational fractions with the same denominators is identically equal to a fraction with the same denominator and with a numerator equal to the sum of the numerators of the fractions being added:

The situation is similar in the case of subtracting fractions with like denominators:

Example 1: Simplify an expression

Solution.

To add or subtract rational fractions with different denominators, you must first reduce the fractions to a common denominator, and then perform operations on the resulting fractions with the same denominators.

Example 2: Simplify an expression

Solution. We have

63. Multiplication and division of rational fractions.

The product of two (and in general any finite number) rational fractions is identically equal to a fraction whose numerator is equal to the product of the numerators, and the denominator is equal to the product of the denominators of the fractions being multiplied:

The quotient of dividing two rational fractions is identically equal to a fraction whose numerator is equal to the product of the numerator of the first fraction and the denominator of the second fraction, and the denominator is the product of the denominator of the first fraction and the numerator of the second fraction:

The formulated rules of multiplication and division also apply to the case of multiplication or division by a polynomial: it is enough to write this polynomial in the form of a fraction with a denominator of 1.

Given the possibility of reducing a rational fraction obtained as a result of multiplying or dividing rational fractions, they usually strive to factorize the numerators and denominators of the original fractions before performing these operations.

Example 1: Perform multiplication

Solution. We have

Using the rule for multiplying fractions, we get:

Example 2: Perform division

Solution. We have

Using the division rule, we get:

64. Raising a rational fraction to a whole power.

To raise a rational fraction to a natural power, you need to raise the numerator and denominator of the fraction separately to this power; the first expression is the numerator, and the second expression is the denominator of the result:

Example 1: Convert to a fraction of power 3.

Solution Solution.

When raising a fraction to a negative integer power, an identity is used that is valid for all values ​​of the variables for which .

Example 2: Convert an expression to a fraction

65. Transformation of rational expressions.

Transforming any rational expression comes down to adding, subtracting, multiplying and dividing rational fractions, as well as raising a fraction to a natural power. Any rational expression can be converted into a fraction, the numerator and denominator of which are whole rational expressions; This, as a rule, is the goal of identical transformations of rational expressions.

Example. Simplify an expression

66. The simplest transformations of arithmetic roots (radicals).

When converting arithmetic korias, their properties are used (see paragraph 35).

Let's look at several examples of using the properties of arithmetic roots for the simplest transformations of radicals. In this case, we will consider all variables to take only non-negative values.

Example 1. Extract the root of a product

Solution. Applying the 1° property, we get:

Example 2. Remove the multiplier from under the root sign

Solution.

This transformation is called removing the factor from under the root sign. The purpose of the transformation is to simplify the radical expression.

Example 3: Simplify.

Solution. By the property of 3° we have. Usually they try to simplify the radical expression, for which they take the factors out of the corium sign. We have

Example 4: Simplify

Solution. Let's transform the expression by introducing a factor under the sign of the root: By property 4° we have

Example 5: Simplify

Solution. By the property of 5°, we have the right to divide the exponent of the root and the exponent of the radical expression by the same natural number. If in the example under consideration we divide the indicated indicators by 3, we get .

Example 6. Simplify expressions:

Solution, a) By property 1° we find that to multiply roots of the same degree, it is enough to multiply the radical expressions and extract the root of the same degree from the result obtained. Means,

b) First of all, we must reduce the radicals to one indicator. According to the property of 5°, we can multiply the exponent of the root and the exponent of the radical expression by the same natural number. Therefore, Next, we now have in the resulting result dividing the exponents of the root and the degree of the radical expression by 3, we get.

>>Math: Converting rational expressions

Converting rational expressions

This paragraph summarizes everything that we, starting from the 7th grade, talked about mathematical language, mathematical symbolism, numbers, variables, powers, polynomials and algebraic fractions. But first, let's take a short excursion into the past.

Remember how things were with the study of numbers and numerical expressions in the lower grades.

And, say, only one label can be attached to a fraction - a rational number.

The situation is similar with algebraic expressions: the first stage of their study is numbers, variables, degrees (“digits”); the second stage of their study is monomials (“natural numbers”); the third stage of their study is polynomials (“integers”); the fourth stage of their study - algebraic fractions
(“rational numbers”). Moreover, each next stage, as it were, absorbs the previous one: for example, numbers, variables, powers are special cases of monomials; monomials - special cases of polynomials; polynomials are special cases of algebraic fractions. By the way, the following terms are sometimes used in algebra: polynomial - integer expression, an algebraic fraction is a fractional expression (this only strengthens the analogy).

Let's continue the above analogy. You know that any numerical expression, after performing all the arithmetic operations included in it, takes on a specific numerical value - a rational number (of course, it can turn out to be a natural number, an integer, or a fraction - it doesn’t matter). Similarly, any algebraic expression composed of numbers and variables using arithmetic operations and raising to natural degree, after performing the transformations takes the form of an algebraic fraction and again, in particular, the result may not be a fraction, but a polynomial or even a monomial). For such expressions in algebra the term rational expression is used.

Example. Prove identity

Solution.
To prove an identity means to establish that for all admissible values ​​of the variables, its left and right sides are identically equal expressions. In algebra, identities are proven in various ways:

1) perform transformations on the left side and ultimately obtain the right side;

2) perform transformations on the right side and ultimately obtain the left side;

3) transform the right and left sides separately and obtain the same expression in both the first and second cases;

4) make up the difference between the left and right sides and, as a result of its transformations, obtain zero.

Which method to choose depends on the specific type identities which you are asked to prove. In this example, it is advisable to choose the first method.

To convert rational expressions, the same procedure is adopted as for converting numerical expressions. This means that first they perform the actions in brackets, then the actions of the second stage (multiplication, division, exponentiation), then the actions of the first stage (addition, subtraction).

Let's carry out transformations based on the rules algorithms that were developed in the previous paragraphs.

As you can see, we were able to transform the left side of the identity being verified to the form of the right side. This means that the identity is proven. However, recall that the identity is valid only for admissible values ​​of the variables. In this example, these are any values ​​of a and b, except those that make the denominators of the fractions zero. This means that any pairs of numbers (a; b) are valid, except those for which at least one of the equalities is satisfied:

2a - b = 0, 2a + b = 0, b = 0.

Mordkovich A. G., Algebra. 8th grade: Textbook. for general education institutions. - 3rd ed., revised. - M.: Mnemosyne, 2001. - 223 p.: ill.

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This lesson will cover basic information about rational expressions and their transformations, as well as examples of transformations of rational expressions. This topic summarizes the topics we have studied so far. Transformations of rational expressions involve addition, subtraction, multiplication, division, exponentiation of algebraic fractions, reduction, factorization, etc. As part of the lesson, we will look at what a rational expression is, and also analyze examples of their transformation.

Subject:Algebraic fractions. Arithmetic operations on algebraic fractions

Lesson:Basic information about rational expressions and their transformations

Definition

Rational expression is an expression consisting of numbers, variables, arithmetic operations and the operation of exponentiation.

Let's look at an example of a rational expression:

Special cases of rational expressions:

1st degree: ;

2. monomial: ;

3. fraction: .

Converting a rational expression is a simplification of a rational expression. The order of actions when transforming rational expressions: first there are operations in brackets, then multiplication (division) operations, and then addition (subtraction) operations.

Let's look at several examples of transforming rational expressions.

Example 1

Solution:

Let's solve this example step by step. The action in parentheses is executed first.

Answer:

Example 2

Solution:

Answer:

Example 3

Solution:

Answer: .

Note: Perhaps, when you saw this example, an idea arose: reduce the fraction before reducing it to a common denominator. Indeed, it is absolutely correct: first it is advisable to simplify the expression as much as possible, and then transform it. Let's try to solve the same example in the second way.

As you can see, the answer turned out to be absolutely similar, but the solution turned out to be somewhat simpler.

In this lesson we looked at rational expressions and their transformations, as well as several specific examples of these transformations.

References

1. Bashmakov M.I. Algebra 8th grade. - M.: Education, 2004.

2. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. and others. Algebra 8. - 5th ed. - M.: Education, 2010.