Topic No. 2.

Converting algebraic expressions

I. Theoretical material

Basic Concepts

Algebraic expression: integer, fractional, rational, irrational.

Scope of definition, valid expression values.

The meaning of an algebraic expression.

Monomial, polynomial.

Abbreviated multiplication formulas.

Factorization, putting the common factor out of brackets.

The main property of a fraction.

Degree, properties of degree.

Kortym, properties of roots.

Transformation of rational and irrational expressions.

An expression made up of numbers and variables using the signs of addition, subtraction, multiplication, division, raising to a rational power, extracting the root and using parentheses is called algebraic.

For example: ;
;
;

;
;
;
.

If an algebraic expression does not contain division into variables and taking the root of variables (in particular, raising to a power with a fractional exponent), then it is called whole.

For example:
;
;
.

If an algebraic expression is composed of numbers and variables using the operations of addition, subtraction, multiplication, exponentiation with a natural exponent and division, and division into expressions with variables is used, then it is called fractional.

For example:
;
.

Integer and fractional expressions are called rational expressions.

For example: ;
;

.

If an algebraic expression involves taking the root of variables (or raising variables to a fractional power), then such an algebraic expression is called irrational.

For example:
;
.

The values ​​of the variables for which the algebraic expression makes sense are called valid variable values.

The set of all possible values ​​of variables is called domain of definition.

The domain of definition of an entire algebraic expression is the set of real numbers.

The domain of definition of a fractional algebraic expression is the set of all real numbers except those that make the denominator zero.

For example: makes sense when
;

makes sense when
, that is, when
.

The domain of definition of an irrational algebraic expression is the set of all real numbers, except those that turn into a negative number the expression under the sign of the root of an even power or under the sign of raising to a fractional power.

For example:
makes sense when
;

makes sense when
, that is, when
.

The numerical value obtained by substituting the permissible values ​​of variables into an algebraic expression is called the value of an algebraic expression.

For example: expression
at
,
takes on the value
.

An algebraic expression containing only numbers, natural powers of variables and their products is called monomial.

For example:
;
;
.

The monomial, written as the product of the numerical factor in the first place and the powers of various variables, is reduced to standard view.

For example:
;
.

The numerical factor of the standard notation of a monomial is called coefficient of the monomial. The sum of the exponents of all variables is called degree of monomial.

When multiplying a monomial by a monomial and raising a monomial to a natural power, we obtain a monomial that must be reduced to standard form.

The sum of monomials is called polynomial.

For example:
; ;
.

If all members of a polynomial are written in standard form and similar members are reduced, then the resulting polynomial of standard form.

For example: .

If there is only one variable in a polynomial, then the largest exponent of this variable is called degree of polynomial.

For example: A polynomial has the fifth degree.

The value of the variable at which the value of the polynomial is zero is called root of the polynomial.

For example: roots of a polynomial
are the numbers 1.5 and 2.

Abbreviated multiplication formulas

Special cases of using abbreviated multiplication formulas

Difference of squares:
or

Squared sum:
or

Squared difference:
or

Sum of cubes:
or

Difference of cubes:
or

Cube of sum:
or

Difference cube:
or

Converting a polynomial into a product of several factors (polynomials or monomials) is called factoring a polynomial.

For example:.

Methods for factoring a polynomial

For example: .

Using abbreviated multiplication formulas.

For example: .

Grouping method. The commutative and associative laws allow the members of a polynomial to be grouped in various ways. One of the methods leads to the fact that the same expression is obtained in brackets, which in turn is taken out of brackets.

For example:.

Any fractional algebraic expression can be written as the quotient of two rational expressions with a variable in the denominator.

For example:
.

A fraction in which the numerator and denominator are rational expressions and the denominator has a variable is called rational fraction.

For example:
;
;
.

If the numerator and denominator of a rational fraction are multiplied or divided by the same nonzero number, monomial, or polynomial, the value of the fraction does not change. This expression is called the main property of a fraction:

.

The action of dividing the numerator and denominator of a fraction by the same number is called reducing a fraction:

.

For example:
;
.

Work n factors, each of which is equal A, Where A is an arbitrary algebraic expression or real number, and n- a natural number, called degreeA :

.

Algebraic expression A called degree basis, number
n – indicator.

For example:
.

It is believed by definition that for any A, not equal to zero:

And
.

If
, That
.

Properties of degree

1.
.

2.
.

3.
.

4.
.

5.
.

If ,
, then the expression n-th degree of which is equal to A, called rootn th degree ofA . It is usually denoted
. At the same time A called radical expression, n called root index.

For example:
;
;
.

Root propertiesnth degree of a

1.
.

2.
,
.

3.
.

4.
.

5.
.

Generalizing the concept of degree and root, we obtain the concept of degree with a rational exponent:

.

In particular,
.

Actions performed with roots

For example: .

II. Practical material

Examples of completing tasks

Example 1. Find the value of the fraction
.

Answer: .

Example 2. Simplify the expression
.

Let's transform the expression in the first brackets:

, If
.

Let's transform the expression in the second brackets:

.

Let's divide the result from the first bracket by the result from the second bracket:

Answer:

Example 3. Simplify the expression:

.

Example 4. Simplify the expression.

Let's transform the first fraction:

.

Let's transform the second fraction:

.

As a result we get:
.

Example 5. Simplify the expression
.

Solution. Let's decide on the following actions:

1)
;

2)
;

3)
;

6)
;

Answer:
.

Example 6. Prove the identity
.

1)
;

2)
;

Example 7. Simplify the expression:

.

Solution. Follow these steps:

;

2)
.

Example 8. Prove the identity
.

Solution. Follow these steps:

1)
;

2)

;

3)
.

Tasks for independent work

1. Simplify the expression:

A)
;

b)
;

2. Factor into:

A)
;

b)
;.Document

Subject No. 5.1. Trigonometric equations I. Theoreticalmaterial Basic concepts Trigonometric equation... using various algebraic and trigonometric formulas and transformations. II. Practical material Examples of completing tasks...

Basic properties of addition and multiplication of numbers.

Commutative property of addition: rearranging the terms does not change the value of the sum. For any numbers a and b the equality is true

Combinative property of addition: in order to add a third number to the sum of two numbers, you can add the sum of the second and third to the first number. For any numbers a, b and c the equality is true

Commutative property of multiplication: rearranging the factors does not change the value of the product. For any numbers a, b and c the equality is true

Combinative property of multiplication: to multiply the product of two numbers by a third number, you can multiply the first number by the product of the second and third.

For any numbers a, b and c the equality is true

Distributive Property: To multiply a number by a sum, you can multiply that number by each term and add the results. For any numbers a, b and c the equality is true

From the commutative and combinative properties of addition it follows: in any sum you can rearrange the terms in any way you like and arbitrarily combine them into groups.

Example 1 Let's calculate the sum 1.23+13.5+4.27.

To do this, it is convenient to combine the first term with the third. We get:

1,23+13,5+4,27=(1,23+4,27)+13,5=5,5+13,5=19.

From the commutative and combinative properties of multiplication it follows: in any product you can rearrange the factors in any way and arbitrarily combine them into groups.

Example 2 Let's find the value of the product 1.8·0.25·64·0.5.

Combining the first factor with the fourth, and the second with the third, we have:

1.8·0.25·64·0.5=(1.8·0.5)·(0.25·64)=0.9·16=14.4.

The distributive property is also true when a number is multiplied by the sum of three or more terms.

For example, for any numbers a, b, c and d the equality is true

a(b+c+d)=ab+ac+ad.

We know that subtraction can be replaced by addition by adding to the minuend the opposite number of the subtrahend:

This allows a numerical expression of the form a-b to be considered the sum of numbers a and -b, a numerical expression of the form a+b-c-d to be considered the sum of numbers a, b, -c, -d, etc. The considered properties of actions are also valid for such sums.

Example 3 Let's find the value of the expression 3.27-6.5-2.5+1.73.

This expression is the sum of the numbers 3.27, -6.5, -2.5 and 1.73. Applying the properties of addition, we get: 3.27-6.5-2.5+1.73=(3.27+1.73)+(-6.5-2.5)=5+(-9) = -4.

Example 4 Let's calculate the product 36·().

The multiplier can be thought of as the sum of the numbers and -. Using the distributive property of multiplication, we obtain:

36()=36·-36·=9-10=-1.

Identities

Definition. Two expressions whose corresponding values ​​are equal for any values ​​of the variables are called identically equal.

Definition. An equality that is true for any values ​​of the variables is called an identity.

Let's find the values ​​of the expressions 3(x+y) and 3x+3y for x=5, y=4:

3(x+y)=3(5+4)=3 9=27,

3x+3y=3·5+3·4=15+12=27.

We got the same result. From the distribution property it follows that, in general, for any values ​​of the variables, the corresponding values ​​of the expressions 3(x+y) and 3x+3y are equal.

Let us now consider the expressions 2x+y and 2xy. When x=1, y=2 they take equal values:

However, you can specify values ​​of x and y such that the values ​​of these expressions are not equal. For example, if x=3, y=4, then

The expressions 3(x+y) and 3x+3y are identically equal, but the expressions 2x+y and 2xy are not identically equal.

The equality 3(x+y)=x+3y, true for any values ​​of x and y, is an identity.

True numerical equalities are also considered identities.

Thus, identities are equalities that express the basic properties of operations on numbers:

a+b=b+a, (a+b)+c=a+(b+c),

ab=ba, (ab)c=a(bc), a(b+c)=ab+ac.

Other examples of identities can be given:

a+0=a, a+(-a)=0, a-b=a+(-b),

a·1=a, a·(-b)=-ab, (-a)(-b)=ab.

Identical transformations of expressions

Replacing one expression with another identically equal expression is called an identical transformation or simply a transformation of an expression.

Identical transformations of expressions with variables are performed based on the properties of operations on numbers.

To find the value of the expression xy-xz for given values ​​of x, y, z, you need to perform three steps. For example, with x=2.3, y=0.8, z=0.2 we get:

xy-xz=2.3·0.8-2.3·0.2=1.84-0.46=1.38.

This result can be obtained by performing only two steps, if you use the expression x(y-z), which is identically equal to the expression xy-xz:

xy-xz=2.3(0.8-0.2)=2.3·0.6=1.38.

We have simplified the calculations by replacing the expression xy-xz with the identically equal expression x(y-z).

Identical transformations of expressions are widely used in calculating the values ​​of expressions and solving other problems. Some identical transformations have already had to be performed, for example, bringing similar terms, opening parentheses. Let us recall the rules for performing these transformations:

to bring similar terms, you need to add their coefficients and multiply the result by the common letter part;

if there is a plus sign before the brackets, then the brackets can be omitted, preserving the sign of each term enclosed in brackets;

If there is a minus sign before the parentheses, then the parentheses can be omitted by changing the sign of each term enclosed in the parentheses.

Example 1 Let us present similar terms in the sum 5x+2x-3x.

Let's use the rule for reducing similar terms:

5x+2x-3x=(5+2-3)x=4x.

This transformation is based on the distributive property of multiplication.

Example 2 Let's open the brackets in the expression 2a+(b-3c).

Using the rule for opening parentheses preceded by a plus sign:

2a+(b-3c)=2a+b-3c.

The transformation carried out is based on the combinatory property of addition.

Example 3 Let's open the brackets in the expression a-(4b-c).

Let's use the rule for opening parentheses preceded by a minus sign:

a-(4b-c)=a-4b+c.

The transformation performed is based on the distributive property of multiplication and the combinatory property of addition. Let's show it. Let us represent the second term -(4b-c) in this expression as a product (-1)(4b-c):

a-(4b-c)=a+(-1)(4b-c).

Applying the specified properties of actions, we get:

a-(4b-c)=a+(-1)(4b-c)=a+(-4b+c)=a-4b+c.

The numbers and expressions that make up the original expression can be replaced by identically equal expressions. Such a transformation of the original expression leads to an expression that is identically equal to it.

For example, in the expression 3+x, the number 3 can be replaced by the sum 1+2, which will result in the expression (1+2)+x, which is identically equal to the original expression. Another example: in the expression 1+a 5, the power a 5 can be replaced by an identically equal product, for example, of the form a·a 4. This will give us the expression 1+a·a 4 .

This transformation is undoubtedly artificial, and is usually a preparation for some further transformations. For example, in the sum 4 x 3 +2 x 2, taking into account the properties of the degree, the term 4 x 3 can be represented as a product 2 x 2 2 x. After this transformation, the original expression will take the form 2 x 2 2 x+2 x 2. Obviously, the terms in the resulting sum have a common factor of 2 x 2, so we can perform the following transformation - bracketing. After it we come to the expression: 2 x 2 (2 x+1) .

Adding and subtracting the same number

Another artificial transformation of an expression is the addition and simultaneous subtraction of the same number or expression. This transformation is identical because it is essentially equivalent to adding zero, and adding zero does not change the value.

Let's look at an example. Let's take the expression x 2 +2·x. If you add one to it and subtract one, this will allow you to perform another identical transformation in the future - square the binomial: x 2 +2 x=x 2 +2 x+1−1=(x+1) 2 −1.

References.

• Algebra: textbook for 7th grade general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 17th ed. - M.: Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
• Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
• Mordkovich A. G. Algebra. 7th grade. In 2 hours. Part 1. Textbook for students of general education institutions / A. G. Mordkovich. - 17th ed., add. - M.: Mnemosyne, 2013. - 175 p.: ill. ISBN 978-5-346-02432-3.

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We often hear this unpleasant phrase: “simplify the expression.” Usually we see some kind of monster like this:

“It’s much simpler,” we say, but such an answer usually doesn’t work.

Now I will teach you not to be afraid of any such tasks.

Moreover, at the end of the lesson, you yourself will simplify this example to (just!) an ordinary number (yes, to hell with these letters).

But before you start this activity, you need to be able to handle fractions And factor polynomials.

Therefore, if you have not done this before, be sure to master the topics “” and “”.

Have you read it? If yes, then you are now ready.

Let's go! (Let's go!)

Basic Expression Simplification Operations

Now let's look at the basic techniques that are used to simplify expressions.

The simplest one is

1. Bringing similar

What are similar? You took this in 7th grade, when letters instead of numbers first appeared in mathematics.

Similar- these are terms (monomials) with the same letter part.

For example, in sum, similar terms are and.

Do you remember?

Give similar- means adding several similar terms to each other and getting one term.

How can we put the letters together? - you ask.

This is very easy to understand if you imagine that the letters are some kind of objects.

For example, a letter is a chair. Then what is the expression equal to?

Two chairs plus three chairs, how many will it be? That's right, chairs: .

Now try this expression: .

To avoid confusion, let different letters represent different objects.

For example, - is (as usual) a chair, and - is a table.

chairs tables chair tables chairs chairs tables

The numbers by which the letters in such terms are multiplied are called coefficients.

For example, in a monomial the coefficient is equal. And in it is equal.

So, the rule for bringing similar ones is:

Examples:

Give similar ones:

Answers:

2. (and similar, since, therefore, these terms have the same letter part).

2. Factorization

This is usually the most important part in simplifying expressions.

After you have given similar ones, most often the resulting expression is needed factorize, that is, presented in the form of a product.

Especially this important in fractions: after all, in order to be able to reduce the fraction, The numerator and denominator must be represented as a product.

You went through the methods of factoring expressions in detail in the topic “”, so here you just have to remember what you learned.

To do this, solve several examples (you need to factorize them)

Examples:

Solutions:

3. Reducing a fraction.

Well, what could be more pleasant than crossing out part of the numerator and denominator and throwing them out of your life?

That's the beauty of downsizing.

It's simple:

If the numerator and denominator contain the same factors, they can be reduced, that is, removed from the fraction.

This rule follows from the basic property of a fraction:

That is, the essence of the reduction operation is that We divide the numerator and denominator of the fraction by the same number (or by the same expression).

To reduce a fraction you need:

1) numerator and denominator factorize

2) if the numerator and denominator contain common factors, they can be crossed out.

Examples:

The principle, I think, is clear?

I would like to draw your attention to one typical mistake when abbreviating. Although this topic is simple, many people do everything wrong, not understanding that reduce- this means divide numerator and denominator are the same number.

No abbreviations if the numerator or denominator is a sum.

For example: we need to simplify.

Some people do this: which is absolutely wrong.

Another example: reduce.

The “smartest” will do this:

Tell me what's wrong here? It would seem: - this is a multiplier, which means it can be reduced.

But no: - this is a factor of only one term in the numerator, but the numerator itself as a whole is not factorized.

Here's another example: .

This expression is factorized, which means you can reduce it, that is, divide the numerator and denominator by, and then by:

You can immediately divide it into:

To avoid such mistakes, remember an easy way to determine whether an expression is factorized:

The arithmetic operation that is performed last when calculating the value of an expression is the “master” operation.

That is, if you substitute some (any) numbers instead of letters and try to calculate the value of the expression, then if the last action is multiplication, then we have a product (the expression is factorized).

If the last action is addition or subtraction, this means that the expression is not factorized (and therefore cannot be reduced).

To reinforce this, solve a few examples yourself:

Examples:

Solutions:

4. Adding and subtracting fractions. Reducing fractions to a common denominator.

Adding and subtracting ordinary fractions is a familiar operation: we look for a common denominator, multiply each fraction by the missing factor and add/subtract the numerators.

Let's remember:

Answers:

1. The denominators and are relatively prime, that is, they do not have common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:

2. Here the common denominator is:

3. Here, first of all, we convert mixed fractions into improper ones, and then according to the usual scheme:

It's a completely different matter if the fractions contain letters, for example:

Let's start with something simple:

a) Denominators do not contain letters

Here everything is the same as with ordinary numerical fractions: we find the common denominator, multiply each fraction by the missing factor and add/subtract the numerators:

Now in the numerator you can give similar ones, if any, and factor them:

Try it yourself:

Answers:

b) Denominators contain letters

Let's remember the principle of finding a common denominator without letters:

· first of all, we determine the common factors;

· then we write out all the common factors one at a time;

· and multiply them by all other non-common factors.

To determine the common factors of the denominators, we first factor them into prime factors:

Let us emphasize the common factors:

Now let’s write out the common factors one at a time and add to them all the non-common (not underlined) factors:

This is the common denominator.

Let's get back to the letters. The denominators are given in exactly the same way:

· factor the denominators;

· determine common (identical) factors;

· write out all common factors once;

· multiply them by all other non-common factors.

So, in order:

1) factor the denominators:

2) determine common (identical) factors:

3) write out all common factors once and multiply them by all other (unemphasized) factors:

So there's a common denominator here. The first fraction must be multiplied by, the second - by:

By the way, there is one trick:

For example: .

We see the same factors in the denominators, only all with different indicators. The common denominator will be:

to a degree

to a degree

to a degree

to a degree.

Let's complicate the task:

How to make fractions have the same denominator?

Let's remember the basic property of a fraction:

Nowhere does it say that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because it's not true!

See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example, . What did you learn?

So, another unshakable rule:

When you reduce fractions to a common denominator, use only the multiplication operation!

But what do you need to multiply by to get?

So multiply by. And multiply by:

We will call expressions that cannot be factorized “elementary factors.”

For example, - this is an elementary factor. - Same. But no: it can be factorized.

What about the expression? Is it elementary?

No, because it can be factorized:

(you already read about factorization in the topic “”).

So, the elementary factors into which you decompose an expression with letters are an analogue of the simple factors into which you decompose numbers. And we will deal with them in the same way.

We see that both denominators have a multiplier. It will go to the common denominator to the degree (remember why?).

The factor is elementary, and they do not have a common factor, which means that the first fraction will simply have to be multiplied by it:

Another example:

Solution:

Before multiplying these denominators in a panic, you need to think about how to factor them? They both represent:

Great! Then:

Another example:

Solution:

As usual, let's factorize the denominators. In the first denominator we simply put it out of brackets; in the second - the difference of squares:

It would seem that there are no common factors. But if you look closely, they are similar... And it’s true:

So let's write:

That is, it turned out like this: inside the bracket we swapped the terms, and at the same time the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.

Now let's bring it to a common denominator:

Got it? Let's check it now.

Tasks for independent solution:

Answers:

5. Multiplication and division of fractions.

Well, the hardest part is over now. And ahead of us is the simplest, but at the same time the most important:

Procedure

What is the procedure for calculating a numerical expression? Remember by calculating the meaning of this expression:

Did you count?

It should work.

So, let me remind you.

The first step is to calculate the degree.

The second is multiplication and division. If there are several multiplications and divisions at the same time, they can be done in any order.

And finally, we perform addition and subtraction. Again, in any order.

But: the expression in brackets is evaluated out of turn!

If several brackets are multiplied or divided by each other, we first calculate the expression in each of the brackets, and then multiply or divide them.

What if there are more brackets inside the brackets? Well, let's think: some expression is written inside the brackets. When calculating an expression, what should you do first? That's right, calculate the brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.

So, the procedure for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):

Okay, it's all simple.

But this is not the same as an expression with letters?

No, it's the same! Only instead of arithmetic operations, you need to do algebraic ones, that is, the actions described in the previous section: bringing similar, adding fractions, reducing fractions, and so on. The only difference will be the action of factoring polynomials (we often use this when working with fractions). Most often, to factorize, you need to use I or simply put the common factor out of brackets.

Usually our goal is to represent an expression as a product or quotient.

For example:

Let's simplify the expression.

1) First, we simplify the expression in brackets. There we have a difference of fractions, and our goal is to present it as a product or quotient. So, we bring the fractions to a common denominator and add:

It is impossible to simplify this expression any further; all the factors here are elementary (do you still remember what this means?).

2) We get:

Multiplying fractions: what could be simpler.

3) Now you can shorten:

Well, that's all. Nothing complicated, right?

Another example:

Simplify the expression.

First, try to solve it yourself, and only then look at the solution.

Solution:

First of all, let's determine the order of actions.

First, let's add the fractions in parentheses, so instead of two fractions we get one.

Then we will do division of fractions. Well, let's add the result with the last fraction.

I will number the steps schematically:

Finally, I will give you two useful tips:

1. If there are similar ones, they must be brought immediately. At whatever point similar ones arise in our country, it is advisable to bring them up immediately.

2. The same applies to reducing fractions: as soon as the opportunity to reduce appears, it must be taken advantage of. The exception is for fractions that you add or subtract: if they now have the same denominators, then the reduction should be left for later.

Here are some tasks for you to solve on your own:

And what was promised at the very beginning:

Answers:

Solutions (brief):

If you have coped with at least the first three examples, then you have mastered the topic.

Now on to learning!

CONVERTING EXPRESSIONS. SUMMARY AND BASIC FORMULAS

Basic simplification operations:

• Bringing similar: to add (reduce) similar terms, you need to add their coefficients and assign the letter part.
• Factorization: putting the common factor out of brackets, applying it, etc.
• Reducing a fraction: The numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, which does not change the value of the fraction.
  1) numerator and denominator factorize
  2) if the numerator and denominator have common factors, they can be crossed out.

  IMPORTANT: only multipliers can be reduced!

• Adding and subtracting fractions:
  ;
• Multiplying and dividing fractions:
  ;

Well, the topic is over. If you are reading these lines, it means you are very cool.

Because only 5% of people are able to master something on their own. And if you read to the end, then you are in this 5%!

Now the most important thing.

You have understood the theory on this topic. And, I repeat, this... this is just super! You are already better than the vast majority of your peers.

The problem is that this may not be enough...

For what?

For successfully passing the Unified State Exam, for entering college on a budget and, MOST IMPORTANTLY, for life.

I won’t convince you of anything, I’ll just say one thing...

People who have received a good education earn much more than those who have not received it. This is statistics.

But this is not the main thing.

The main thing is that they are MORE HAPPY (there are such studies). Perhaps because many more opportunities open up before them and life becomes brighter? Don't know...

But think for yourself...

What does it take to be sure to be better than others on the Unified State Exam and ultimately be... happier?

GAIN YOUR HAND BY SOLVING PROBLEMS ON THIS TOPIC.

You won't be asked for theory during the exam.

You will need solve problems against time.

And, if you haven’t solved them (A LOT!), you’ll definitely make a stupid mistake somewhere or simply won’t have time.

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Numerical and algebraic expressions. Converting Expressions.

What is an expression in mathematics? Why do we need expression conversions?

The question, as they say, is interesting... The fact is that these concepts are the basis of all mathematics. All mathematics consists of expressions and their transformations. Not very clear? Let me explain.

Let's say you have an evil example in front of you. Very big and very complex. Let's say you're good at math and aren't afraid of anything! Can you give an answer right away?

You will have to decide this example. Consistently, step by step, this example simplify. According to certain rules, of course. Those. do expression conversion. The more successfully you carry out these transformations, the stronger you are in mathematics. If you don't know how to do the right transformations, you won't be able to do them in math. Nothing...

To avoid such an uncomfortable future (or present...), it doesn’t hurt to understand this topic.)

First, let's find out what is an expression in mathematics. What's happened numeric expression and what is algebraic expression.

What is an expression in mathematics?

Expression in mathematics- this is a very broad concept. Almost everything we deal with in mathematics is a set of mathematical expressions. Any examples, formulas, fractions, equations, and so on - it all consists of mathematical expressions.

3+2 is a mathematical expression. s 2 - d 2- this is also a mathematical expression. Both a healthy fraction and even one number are all mathematical expressions. For example, the equation is:

5x + 2 = 12

consists of two mathematical expressions connected by an equal sign. One expression is on the left, the other on the right.

In general, the term " mathematical expression"is used, most often, to avoid humming. They will ask you what an ordinary fraction is, for example? And how to answer?!

First answer: "This is... mmmmmm... such a thing... in which... Can I write a fraction better? Which one do you want?"

The second answer: “An ordinary fraction is (cheerfully and joyfully!) mathematical expression , which consists of a numerator and a denominator!"

The second option will be somehow more impressive, right?)

This is the purpose of the phrase " mathematical expression "very good. Both correct and solid. But for practical use you need to have a good understanding of specific types of expressions in mathematics .

The specific type is another matter. This It's a completely different matter! Each type of mathematical expression has mine a set of rules and techniques that must be used when making a decision. For working with fractions - one set. For working with trigonometric expressions - the second one. For working with logarithms - the third. And so on. Somewhere these rules coincide, somewhere they differ sharply. But don't be afraid of these scary words. We will master logarithms, trigonometry and other mysterious things in the appropriate sections.

Here we will master (or - repeat, depending on who...) two main types of mathematical expressions. Numerical expressions and algebraic expressions.

Numeric expressions.

What's happened numeric expression? This is a very simple concept. The name itself hints that this is an expression with numbers. Yes, that's how it is. A mathematical expression made up of numbers, parentheses and arithmetic symbols is called a numerical expression.

7-3 is a numerical expression.

(8+3.2) 5.4 is also a numerical expression.

And this monster:

also a numerical expression, yes...

An ordinary number, a fraction, any example of calculation without X's and other letters - all these are numerical expressions.

Main sign numerical expressions - in it no letters. None. Only numbers and mathematical symbols (if necessary). It's simple, right?

And what can you do with numerical expressions? Numeric expressions can usually be counted. To do this, it happens that you have to open the brackets, change signs, abbreviate, swap terms - i.e. do expression conversions. But more on that below.

Here we will deal with such a funny case when with a numerical expression you don't need to do anything. Well, nothing at all! This pleasant operation - do nothing)- is executed when the expression doesn't make sense.

When does a numerical expression make no sense?

It’s clear that if we see some kind of abracadabra in front of us, like

then we won’t do anything. Because it’s not clear what to do about it. Some kind of nonsense. Maybe count the number of pluses...

But there are outwardly quite decent expressions. For example this:

(2+3) : (16 - 2 8)

However, this expression also doesn't make sense! For the simple reason that in the second brackets - if you count - you get zero. But you can’t divide by zero! This is a forbidden operation in mathematics. Therefore, there is no need to do anything with this expression either. For any task with such an expression, the answer will always be the same: "The expression has no meaning!"

To give such an answer, of course, I had to calculate what would be in brackets. And sometimes there’s a lot of stuff in parentheses... Well, there’s nothing you can do about it.

There are not so many forbidden operations in mathematics. There is only one in this topic. Division by zero. Additional restrictions arising in roots and logarithms are discussed in the corresponding topics.

So, an idea of ​​what it is numeric expression- received. Concept the numeric expression doesn't make sense- realized. Let's move on.

Algebraic expressions.

If letters appear in a numerical expression, this expression becomes... The expression becomes... Yes! It becomes algebraic expression. For example:

5a 2; 3x-2y; 3(z-2); 3.4m/n; x 2 +4x-4; (a+b) 2; ...

Such expressions are also called literal expressions. Or expressions with variables. It's practically the same thing. Expression 5a +c, for example, both literal and algebraic, and an expression with variables.

Concept algebraic expression - broader than numeric. It includes and all numerical expressions. Those. a numerical expression is also an algebraic expression, only without letters. Every herring is a fish, but not every fish is a herring...)

Why alphabetic- It's clear. Well, since there are letters... Phrase expression with variables It’s also not very puzzling. If you understand that numbers are hidden under the letters. All sorts of numbers can be hidden under letters... And 5, and -18, and anything else. That is, a letter can be replace for different numbers. That's why the letters are called variables.

In expression y+5, For example, at- variable value. Or they just say " variable", without the word "magnitude". Unlike five, which is a constant value. Or simply - constant.

Term algebraic expression means that to work with this expression you need to use laws and rules algebra. If arithmetic works with specific numbers, then algebra- with all the numbers at once. A simple example for clarification.

In arithmetic we can write that

But if we write such an equality through algebraic expressions:

a + b = b + a

we'll decide right away All questions. For all numbers in one fell swoop. For everything infinite. Because under the letters A And b implied All numbers. And not only numbers, but even other mathematical expressions. This is how algebra works.

When does an algebraic expression not make sense?

Everything about the numerical expression is clear. You can't divide by zero there. And with letters, is it possible to find out what we are dividing by?!

Let's take for example this expression with variables:

2: (A - 5)

Does it make sense? Who knows? A- any number...

Any, any... But there is one meaning A, for which this expression exactly doesn't make sense! And what is this number? Yes! This is 5! If the variable A replace (they say “substitute”) with the number 5, in brackets you get zero. Which cannot be divided. So it turns out that our expression doesn't make sense, If a = 5. But for other values A does it make sense? Can you substitute other numbers?

Certainly. In such cases they simply say that the expression

2: (A - 5)

makes sense for any values A, except a = 5 .

The whole set of numbers that Can substituting into a given expression is called range of acceptable values this expression.

As you can see, there is nothing tricky. Let's look at the expression with variables and figure out: at what value of the variable is the forbidden operation (division by zero) obtained?

And then be sure to look at the task question. What are they asking?

doesn't make sense, our forbidden meaning will be the answer.

If you ask at what value of a variable the expression makes sense(feel the difference!), the answer will be all other numbers except for the forbidden.

Why do we need the meaning of the expression? He is there, he is not... What's the difference?! The point is that this concept becomes very important in high school. Extremely important! This is the basis for such solid concepts as the domain of acceptable values ​​or the domain of a function. Without this, you will not be able to solve serious equations or inequalities at all. Like this.

Converting Expressions. Identity transformations.

We were introduced to numerical and algebraic expressions. We understood what the phrase “the expression has no meaning” means. Now we need to figure out what it is transformation of expressions. The answer is simple, to the point of disgrace.) This is any action with an expression. That's all. You have been doing these transformations since first grade.

Let's take the cool numerical expression 3+5. How can it be converted? Yes, very simple! Calculate:

This calculation will be the transformation of the expression. You can write the same expression differently:

Here we didn’t count anything at all. Just wrote down the expression in a different form. This will also be a transformation of the expression. You can write it like this:

And this too is a transformation of an expression. You can make as many such transformations as you want.

Any action on expression any writing it in another form is called transforming the expression. And that's all. It's very simple. But there is one thing here very important rule. So important that it can safely be called main rule all mathematics. Breaking this rule inevitably leads to errors. Are we getting into it?)

Let's say we transformed our expression haphazardly, like this:

Conversion? Certainly. We wrote the expression in a different form, what’s wrong here?

It's not like that.) The point is that transformations "at random" are not interested in mathematics at all.) All mathematics is built on transformations in which the appearance changes, but the essence of the expression does not change. Three plus five can be written in any form, but it must be eight.

Transformations, expressions that do not change the essence are called identical.

Exactly identity transformations and allow us, step by step, to transform a complex example into a simple expression, while maintaining the essence of the example. If we make a mistake in the chain of transformations, we make a NOT identical transformation, then we will decide another example. With other answers that are not related to the correct ones.)

This is the main rule for solving any tasks: maintaining the identity of transformations.

I gave an example with the numerical expression 3+5 for clarity. In algebraic expressions, identity transformations are given by formulas and rules. Let's say in algebra there is a formula:

a(b+c) = ab + ac

This means that in any example we can instead of the expression a(b+c) feel free to write an expression ab + ac. And vice versa. This identical transformation. Mathematics gives us a choice between these two expressions. And which one to write depends on the specific example.

Another example. One of the most important and necessary transformations is the basic property of a fraction. You can see more details at the link, but here I’ll just remind you of the rule: If the numerator and denominator of a fraction are multiplied (divided) by the same number, or an expression that is not equal to zero, the fraction will not change. Here is an example of identity transformations using this property:

As you probably guessed, this chain can be continued indefinitely...) A very important property. It is this that allows you to turn all sorts of example monsters into white and fluffy.)

There are many formulas defining identical transformations. But the most important ones are quite a reasonable number. One of the basic transformations is factorization. It is used in all mathematics - from elementary to advanced. Let's start with him. In the next lesson.)

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