How to simplify an algebraic expression. How to simplify a mathematical expression

Expressions, expression conversion

Power expressions (expressions with powers) and their transformation

In this article we will talk about converting expressions with powers. First, we will focus on transformations that are performed with expressions of any kind, including power expressions, such as opening parentheses and bringing similar terms. And then we will analyze the transformations inherent specifically in expressions with degrees: working with the base and exponent, using the properties of degrees, etc.

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What are power expressions?

The term “power expressions” is almost never used school textbooks mathematics, but it appears quite often in collections of problems, especially those intended for preparation for the Unified State Exam and the Unified State Exam, for example. After analyzing the tasks in which it is necessary to perform any actions with power expressions, it becomes clear that power expressions are understood as expressions containing powers in their entries. Therefore, you can accept the following definition for yourself:

Definition.

Power expressions are expressions containing degrees.

Let's give examples of power expressions. Moreover, we will present them according to how the development of views on from a degree with a natural exponent to a degree with a real exponent occurs.

As is known, first one gets acquainted with the power of a number with a natural exponent; at this stage, the first simplest power expressions of the type 3 2, 7 5 +1, (2+1) 5, (−0.1) 4, 3 a 2 appear −a+a 2 , x 3−1 , (a 2) 3 etc.

A little later, the power of a number with an integer exponent is studied, which leads to the appearance of power expressions with integers negative powers, like the following: 3 −2 , , a −2 +2 b −3 +c 2 .

In high school they return to degrees. There a degree with a rational exponent is introduced, which entails the appearance of the corresponding power expressions: , , and so on. Finally, degrees with irrational exponents and expressions containing them are considered: , .

The matter is not limited to the listed power expressions: further the variable penetrates into the exponent, and, for example, the following expressions arise: 2 x 2 +1 or . And after getting acquainted with , expressions with powers and logarithms begin to appear, for example, x 2·lgx −5·x lgx.

So, we have dealt with the question of what power expressions represent. Next we will learn to transform them.

Main types of transformations of power expressions

With power expressions, you can perform any of the basic identity transformations of expressions. For example, you can expand the brackets, replace numeric expressions their values, give similar terms, etc. Naturally, in this case, it is necessary to follow the accepted procedure for performing actions. Let's give examples.

Example.

Calculate the value of the power expression 2 3 ·(4 2 −12) .

Solution.

According to the order of execution of actions, first perform the actions in brackets. There, firstly, we replace the power 4 2 with its value 16 (if necessary, see), and secondly, we calculate the difference 16−12=4. We have 2 3 ·(4 2 −12)=2 3 ·(16−12)=2 3 ·4.

In the resulting expression, we replace the power 2 3 with its value 8, after which we calculate the product 8·4=32. This is the desired value.

So, 2 3 ·(4 2 −12)=2 3 ·(16−12)=2 3 ·4=8·4=32.

Answer:

2 3 ·(4 2 −12)=32.

Example.

Simplify expressions with powers 3 a 4 b −7 −1+2 a 4 b −7.

Solution.

Obviously, this expression contains similar terms 3·a 4 ·b −7 and 2·a 4 ·b −7 , and we can present them: .

Answer:

3 a 4 b −7 −1+2 a 4 b −7 =5 a 4 b −7 −1.

Example.

Express an expression with powers as a product.

Solution.

You can cope with the task by representing the number 9 as a power of 3 2 and then using the formula for abbreviated multiplication - difference of squares:

Answer:

There are also a number of identical transformations inherent specifically in power expressions. We will analyze them further.

Working with base and exponent

There are degrees whose base and/or exponent are not just numbers or variables, but some expressions. As an example, we give the entries (2+0.3·7) 5−3.7 and (a·(a+1)−a 2) 2·(x+1) .

When working with such expressions, you can replace both the expression in the base of the degree and the expression in the exponent with an identically equal expression in the ODZ of its variables. In other words, according to the rules known to us, we can separately transform the base of the degree and separately the exponent. It is clear that as a result of this transformation, an expression will be obtained that is identically equal to the original one.

Such transformations allow us to simplify expressions with powers or achieve other goals we need. For example, in the power expression mentioned above (2+0.3 7) 5−3.7, you can perform operations with the numbers in the base and exponent, which will allow you to move to the power 4.1 1.3. And after opening the brackets and bringing similar terms to the base of the degree (a·(a+1)−a 2) 2·(x+1) we obtain a power expression more simple type a 2·(x+1) .

Using Degree Properties

One of the main tools for transforming expressions with powers is equalities that reflect . Let us recall the main ones. For any positive numbers a and b and arbitrary real numbers r and s, the following properties of powers are true:

• a r ·a s =a r+s ;
• a r:a s =a r−s ;
• (a·b) r =a r ·b r ;
• (a:b) r =a r:b r ;
• (a r) s =a r·s .

Note that for natural, integer, and positive exponents, the restrictions on the numbers a and b may not be so strict. For example, for natural numbers m and n the equality a m ·a n =a m+n is true not only for positive a, but also for negative a, and for a=0.

At school, the main focus when transforming power expressions is on the ability to choose the appropriate property and apply it correctly. In this case, the bases of degrees are usually positive, which allows the properties of degrees to be used without restrictions. The same applies to the transformation of expressions containing variables in the bases of powers - the range of permissible values ​​of the variables is usually such that the bases on it take only positive values, which allows you to freely use the properties of degrees. In general, you need to constantly ask yourself whether it is possible to use any property of degrees in this case, because inaccurate use of properties can lead to a narrowing of the educational value and other troubles. These points are discussed in detail and with examples in the article transformation of expressions using properties of powers. Here we will limit ourselves to considering a few simple examples.

Example.

Express the expression a 2.5 ·(a 2) −3:a −5.5 as a power with base a.

Solution.

First, we transform the second factor (a 2) −3 using the property of raising a power to a power: (a 2) −3 =a 2·(−3) =a −6. The original power expression will take the form a 2.5 ·a −6:a −5.5. Obviously, it remains to use the properties of multiplication and division of powers with the same base, we have
a 2.5 ·a −6:a −5.5 =
a 2.5−6:a −5.5 =a −3.5:a −5.5 =
a −3.5−(−5.5) =a 2 .

Answer:

a 2.5 ·(a 2) −3:a −5.5 =a 2.

Properties of powers when transforming power expressions are used both from left to right and from right to left.

Example.

Find the value of the power expression.

Solution.

The equality (a·b) r =a r ·b r, applied from right to left, allows us to move from the original expression to a product of the form and further. And when multiplying powers with the same bases, the exponents add up: .

It was possible to transform the original expression in another way:

Answer:

.

Example.

Given the power expression a 1.5 −a 0.5 −6, introduce a new variable t=a 0.5.

Solution.

The degree a 1.5 can be represented as a 0.5 3 and then, based on the property of the degree to the degree (a r) s =a r s, applied from right to left, transform it to the form (a 0.5) 3. Thus, a 1.5 −a 0.5 −6=(a 0.5) 3 −a 0.5 −6. Now it’s easy to introduce a new variable t=a 0.5, we get t 3 −t−6.

Answer:

t 3 −t−6 .

Converting fractions containing powers

Power expressions can contain or represent fractions with powers. Any of the basic transformations of fractions that are inherent in fractions of any kind are fully applicable to such fractions. That is, fractions that contain powers can be reduced, reduced to a new denominator, worked separately with their numerator and separately with the denominator, etc. To illustrate these words, consider solutions to several examples.

Example.

Simplify power expression .

Solution.

This power expression is a fraction. Let's work with its numerator and denominator. In the numerator we open the brackets and simplify the resulting expression using the properties of powers, and in the denominator we present similar terms:

And let’s also change the sign of the denominator by placing a minus in front of the fraction: .

Answer:

.

Reducing fractions containing powers to a new denominator is carried out similarly to reducing rational fractions to a new denominator. In this case, an additional factor is also found and the numerator and denominator of the fraction are multiplied by it. When performing this action, it is worth remembering that reduction to a new denominator can lead to a narrowing of the VA. To prevent this from happening, it is necessary that the additional factor does not go to zero for any values ​​of the variables from the ODZ variables for the original expression.

Example.

Reduce the fractions to a new denominator: a) to denominator a, b) to the denominator.

Solution.

a) In this case, it is quite easy to figure out which additional multiplier helps to achieve the desired result. This is a multiplier of a 0.3, since a 0.7 ·a 0.3 =a 0.7+0.3 =a. Note that in the range of permissible values ​​of the variable a (this is the set of all positive real numbers), the power of a 0.3 does not vanish, therefore, we have the right to multiply the numerator and denominator of a given fraction by this additional factor:

b) Taking a closer look at the denominator, you will find that

and multiplying this expression by will give the sum of cubes and , that is, . And this is the new denominator to which we need to reduce the original fraction.

This is how we found an additional factor. In the range of permissible values ​​of the variables x and y, the expression does not vanish, therefore, we can multiply the numerator and denominator of the fraction by it:

Answer:

A) , b) .

There is also nothing new in reducing fractions containing powers: the numerator and denominator are represented as a number of factors, and the same factors of the numerator and denominator are reduced.

Example.

Reduce the fraction: a) , b) .

Solution.

a) Firstly, the numerator and denominator can be reduced by the numbers 30 and 45, which is equal to 15. It is also obviously possible to perform a reduction by x 0.5 +1 and by . Here's what we have:

b) In this case, identical factors in the numerator and denominator are not immediately visible. To obtain them, you will have to perform preliminary transformations. In this case, they consist in factoring the denominator using the difference of squares formula:

Answer:

A)

b) .

Converting fractions to a new denominator and reducing fractions are mainly used to do things with fractions. Actions are performed according to known rules. When adding (subtracting) fractions, they are reduced to a common denominator, after which the numerators are added (subtracted), but the denominator remains the same. The result is a fraction whose numerator is the product of the numerators, and the denominator is the product of the denominators. Division by a fraction is multiplication by its inverse.

Example.

Follow the steps .

Solution.

First, we subtract the fractions in parentheses. To do this, we bring them to a common denominator, which is , after which we subtract the numerators:

Now we multiply the fractions:

Obviously, it is possible to reduce by a power of x 1/2, after which we have .

You can also simplify the power expression in the denominator by using the difference of squares formula: .

Answer:

Example.

Simplify the Power Expression .

Solution.

Obviously, this fraction can be reduced by (x 2.7 +1) 2, this gives the fraction . It is clear that something else needs to be done with the powers of X. To do this, we transform the resulting fraction into a product. This gives us the opportunity to take advantage of the property of dividing powers with the same bases: . And at the end of the process we move from the last product to the fraction.

Answer:

.

And let us also add that it is possible, and in many cases desirable, to transfer factors with negative exponents from the numerator to the denominator or from the denominator to the numerator, changing the sign of the exponent. Such transformations often simplify further actions. For example, a power expression can be replaced by .

Converting expressions with roots and powers

Often, in expressions in which some transformations are required, roots with fractional exponents are also present along with powers. To convert such an expression to the right type, in most cases it is enough to go only to roots or only to powers. But since it is more convenient to work with powers, they usually move from roots to powers. However, it is advisable to carry out such a transition when the ODZ of variables for the original expression allows you to replace the roots with powers without the need to refer to the module or split the ODZ into several intervals (we discussed this in detail in the article transition from roots to powers and back After getting acquainted with the degree with a rational exponent a degree with an irrational exponent is introduced, which allows us to talk about a degree with an arbitrary real exponent. At this stage, it begins to be studied at school. exponential function , which is analytically given by a power, the base of which is a number, and the exponent is a variable. So we are faced with power expressions containing numbers in the base of the power, and in the exponent - expressions with variables, and naturally the need arises to perform transformations of such expressions.

It should be said that the transformation of expressions of the indicated type usually has to be performed when solving exponential equations And exponential inequalities, and these conversions are quite simple. In the overwhelming majority of cases, they are based on the properties of the degree and are aimed, for the most part, at introducing a new variable in the future. The equation will allow us to demonstrate them 5 2 x+1 −3 5 x 7 x −14 7 2 x−1 =0.

Firstly, powers, in the exponents of which is the sum of a certain variable (or expression with variables) and a number, are replaced by products. This applies to the first and last terms of the expression on the left side:
5 2 x 5 1 −3 5 x 7 x −14 7 2 x 7 −1 =0,
5 5 2 x −3 5 x 7 x −2 7 2 x =0.

Next, both sides of the equality are divided by the expression 7 2 x, which on the ODZ of the variable x for the original equation takes only positive values ​​(this is a standard technique for solving equations of this type, we are not talking about it now, so focus on subsequent transformations of expressions with powers ):

Now we can cancel fractions with powers, which gives .

Finally, the ratio of powers with the same exponents is replaced by powers of relations, resulting in the equation , which is equivalent . The transformations made allow us to introduce a new variable, which reduces the solution to the original exponential equation to solving a quadratic equation

Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ... discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.

From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus of application variable units the measurement has either not yet been developed or has not been applied to Zeno's aporia. The use of our ordinary logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.

If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”

How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But it is not complete solution Problems. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia of Zeno tells about a flying arrow:

A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to the car, you need two photographs taken from different points space at one point in time, but it is impossible to determine the fact of movement from them (naturally, additional data is still needed for calculations, trigonometry will help you). What I want to point out Special attention, is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.

Wednesday, July 4, 2018

The differences between set and multiset are described very well on Wikipedia. Let's see.

As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.

Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.

No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.

We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.

First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to assure us that the banknotes of the same denomination have different numbers bills, which means they cannot be considered identical elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...

And now I have the most interest Ask: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.

Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.

To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."

Sunday, March 18, 2018

The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.

Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.

Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.

1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.

2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.

3. Convert individual graphic symbols into numbers. This is not a mathematical operation.

4. Add the resulting numbers. Now this is mathematics.

The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.

From a mathematical point of view, it does not matter in which number system we write a number. So, in different number systems the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. WITH a large number 12345 I don’t want to fool my head, let’s look at the number 26 from the article about . Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.

As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of ​​a rectangle in meters and centimeters, you would get completely different results.

Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.

The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.

What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.

Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?

Female... The halo on top and the arrow down are male.

If such a work of design art flashes before your eyes several times a day,

Then it’s not surprising that you suddenly find a strange icon in your car:

Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.

1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.

I. Expressions in which numbers, arithmetic symbols and parentheses can be used along with letters are called algebraic expressions.

Examples of algebraic expressions:

2m -n; 3 · (2a + b); 0.24x; 0.3a -b · (4a + 2b); a 2 – 2ab;

Since a letter in an algebraic expression can be replaced by some different numbers, then the letter is called a variable, and the algebraic expression itself is called an expression with a variable.

II. If in an algebraic expression the letters (variables) are replaced by their values ​​and the specified actions are performed, then the resulting number is called the value of the algebraic expression.

Examples.

Find the meaning of the expression:

1) a + 2b -c with a = -2; b = 10; c = -3.5.

2) |x| + |y| -|z| at x = -8; y = -5; z = 6..

Solution

— 2+ 2 · 10- (-3,5) = -2 + 20 +3,5 = 18 + 3,5 = 21,5.

1) a + 2b -c with a = -2; b = 10; c = -3.5. Instead of variables, let's substitute their values. We get: 2) |x| + |y| -|z| at x = -8; y = -5; z = 6. Substitute the indicated values. Remember that the module negative number is equal to its opposite number, and the module positive number

|-8| + |-5| -|6| = 8 + 5 -6 = 7.

equal to this number itself. We get: III.

The values ​​of the letter (variable) for which the algebraic expression makes sense are called the permissible values ​​of the letter (variable).

Solution. Examples.

For what values ​​of the variable does the expression make no sense?

We know that you cannot divide by zero, therefore, each of these expressions will not make sense given the value of the letter (variable) that turns the denominator of the fraction to zero!

In example 1) this value is a = 0. Indeed, if you substitute 0 instead of a, then you will need to divide the number 6 by 0, but this cannot be done. Answer: expression 1) does not make sense when a = 0.

In example 4) the denominator is 5 -|x| = 0 for |x| = 5. And since |5| = 5 and |-5| = 5, then you cannot take x = 5 and x = -5. Answer: expression 4) does not make sense at x = -5 and at x = 5.
IV. Two expressions are said to be identically equal if, for any admissible values ​​of the variables, the corresponding values ​​of these expressions are equal.

Example: 5 (a – b) and 5a – 5b are also equal, since the equality 5 (a – b) = 5a – 5b will be true for any values ​​of a and b. The equality 5 (a – b) = 5a – 5b is an identity.

Identity is an equality that is valid for all permissible values ​​of the variables included in it. Examples of identities already known to you are, for example, the properties of addition and multiplication, and the distributive property.

Replacing one expression with another identically equal expression is called an identity transformation or simply a transformation of an expression. Identical transformations of expressions with variables are performed based on the properties of operations on numbers.

Examples.

a) convert the expression to identically equal using the distributive property of multiplication:

1) 10·(1.2x + 2.3y); 2) 1.5·(a -2b + 4c); 3) a·(6m -2n + k).

2) |x| + |y| -|z| at x = -8; y = -5; z = 6.. Let us recall the distributive property (law) of multiplication:

(a+b)c=ac+bc(distributive law of multiplication relative to addition: in order to multiply the sum of two numbers by a third number, you can multiply each term by this number and add the resulting results).
(a-b) c=a c-b c(distributive law of multiplication relative to subtraction: in order to multiply the difference of two numbers by a third number, you can multiply the minuend and subtract by this number separately and subtract the second from the first result).

1) 10·(1.2x + 2.3y) = 10 · 1.2x + 10 · 2.3y = 12x + 23y.

2) 1.5·(a -2b + 4c) = 1.5a -3b + 6c.

3) a·(6m -2n + k) = 6am -2an +ak.

b) transform the expression into identically equal, using the commutative and associative properties (laws) of addition:

4) x + 4.5 +2x + 6.5; 5) (3a + 2.1) + 7.8; 6) 5.4s -3 -2.5 -2.3s.

Solution. Let's apply the laws (properties) of addition:

a+b=b+a(commutative: rearranging the terms does not change the sum).
(a+b)+c=a+(b+c)(combinative: in order to add a third number to the sum of two terms, you can add the sum of the second and third to the first number).

4) x + 4.5 +2x + 6.5 = (x + 2x) + (4.5 + 6.5) = 3x + 11.

5) (3a + 2.1) + 7.8 = 3a + (2.1 + 7.8) = 3a + 9.9.

6) 6) 5.4s -3 -2.5 -2.3s = (5.4s -2.3s) + (-3 -2.5) = 3.1s -5.5.

V) Convert the expression to identically equal using the commutative and associative properties (laws) of multiplication:

7) 4 · X · (-2,5); 8) -3,5 · 2у · (-1); 9) 3a · (-3) · 2s.

Solution. Let's apply the laws (properties) of multiplication:

a·b=b·a(commutative: rearranging the factors does not change the product).
(a b) c=a (b c)(combinative: 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).

7) 4 · X · (-2,5) = -4 · 2,5 · x = -10x.

8) -3,5 · 2у · (-1) = 7у.

9) 3a · (-3) · 2c = -18ac.

If an algebraic expression is given in the form of a reducible fraction, then using the rule for reducing a fraction it can be simplified, i.e. replace it with an identical, simpler expression.

Examples.

Solution. Simplify by using fraction reduction. To reduce a fraction means to divide its numerator and denominator by the same number (expression) other than zero. Fraction 10) will be reduced by 3b ; fraction 11) reduce by A and fraction 12) will be reduced by 7n

. We get:

Algebraic expressions are used to create formulas. A formula is an algebraic expression written as an equality and expressing the relationship between two or more variables. Example: path formula you know s=v t

(s - distance traveled, v - speed, t - time). Remember what other formulas you know.

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§ 1 The concept of simplifying a literal expression

In this lesson, we will get acquainted with the concept of “similar terms” and, using examples, we will learn how to perform the reduction of similar terms, thus simplifying literal expressions. Let’s find out the meaning of the concept “simplification”. The word “simplification” is derived from the word “simplify”. To simplify means to make simple, simpler. Therefore, to simplify a literal expression is to make it shorter, with minimum quantity

actions.

Consider the expression 9x + 4x. This is a literal expression that is a sum. The terms here are presented as products of a number and a letter. The numerical factor of such terms is called a coefficient. In this expression, the coefficients will be the numbers 9 and 4. Please note that the factor represented by the letter is the same in both terms of this sum.

Let us recall the distributive law of multiplication:

To multiply a sum by a number, you can multiply each term by that number and add the resulting products. IN general view

written as follows: (a + b) ∙ c = ac + bc.

This law is true in both directions ac + bc = (a + b) ∙ c

Let's apply it to our literal expression: the sum of the products of 9x and 4x is equal to a product whose first factor is equal to the sum of 9 and 4, the second factor is x.

9 + 4 = 13, that's 13x.

Instead of three actions in the expression, there is only one action left - multiplication. This means that we have made our literal expression simpler, i.e. simplified it.

§ 2 Reduction of similar terms

The terms 9x and 4x differ only in their coefficients - such terms are called similar. The letter part of similar terms is the same. Similar terms also include numbers and equal terms.

For example, in the expression 9a + 12 - 15 similar terms will be the numbers 12 and -15, and in the sum of the product of 12 and 6a, the number 14 and the product of 12 and 6a (12 ∙ 6a + 14 + 12 ∙ 6a) the equal terms represented by the product of 12 and 6a.

It is important to note that terms whose coefficients are equal, but whose letter factors are different, are not similar, although it is sometimes useful to apply the distributive law of multiplication to them, for example, the sum of the products 5x and 5y is equal to the product of the number 5 and the sum of x and y

5x + 5y = 5(x + y).

Let's simplify the expression -9a + 15a - 4 + 10.

Similar terms in this case are terms -9a and 15a, since they differ only in their coefficients. Their letter multiplier is the same, and the terms -4 and 10 are also similar, since they are numbers. Add up similar terms:

9a + 15a - 4 + 10

9a + 15a = 6a;

We get: 6a + 6.

By simplifying the expression, we found the sums of similar terms; in mathematics this is called reduction of similar terms.

If adding such terms is difficult, you can come up with words for them and add objects.

For example, consider the expression:

For each letter we take our own object: b-apple, c-pear, then we get: 2 apples minus 5 pears plus 8 pears.

Can we subtract pears from apples? Of course not. But we can add 8 pears to minus 5 pears.

Let us present similar terms -5 pears + 8 pears. Similar terms have the same letter part, so when bringing similar terms it is enough to add the coefficients and add the letter part to the result:

(-5 + 8) pears - you get 3 pears.

Returning to our literal expression, we have -5 s + 8 s = 3 s. Thus, after bringing similar terms, we obtain the expression 2b + 3c.

So, in this lesson you became acquainted with the concept of “similar terms” and learned how to simplify letter expressions by reducing similar terms.

List of used literature:

1. Mathematics. Grade 6: lesson plans for I.I.’s textbook. Zubareva, A.G. Mordkovich // author-compiler L.A. Topilina. Mnemosyne 2009.
2. Mathematics. 6th grade: textbook for students educational institutions. I.I. Zubareva, A.G. Mordkovich. - M.: Mnemosyne, 2013.
3. Mathematics. 6th grade: textbook for general education institutions/G.V. Dorofeev, I.F. Sharygin, S.B. Suvorov and others/edited by G.V. Dorofeeva, I.F. Sharygina; Russian Academy of Sciences, Russian Academy of Education. M.: “Enlightenment”, 2010.
4. Mathematics. 6th grade: study for general educational institutions/N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwartzburd. – M.: Mnemosyna, 2013.
5. Mathematics. 6th grade: textbook/G.K. Muravin, O.V. Muravina. – M.: Bustard, 2014.

Images used:

Some algebraic examples alone can terrify schoolchildren. Long expressions are not only intimidating, but also make calculations very difficult. Trying to immediately understand what follows what, it won’t take long to get confused. It is for this reason that mathematicians always try to simplify a “terrible” problem as much as possible and only then begin to solve it. Oddly enough, this trick significantly speeds up the work process.

Simplification is one of the fundamental points in algebra. If you can still do without it in simple problems, then more difficult to calculate examples may turn out to be too tough. This is where these skills come in handy! Moreover, complex mathematical knowledge is not required: it will be enough just to remember and learn to apply in practice a few basic techniques and formulas.

Regardless of the complexity of the calculations, when solving any expression it is important follow the order of performing operations with numbers:

1. brackets;
2. exponentiation;
3. multiplication;
4. division;
5. addition;
6. subtraction.

The last two points can be easily swapped and this will not affect the result in any way. But adding two adjacent numbers when there is a multiplication sign next to one of them is absolutely forbidden! The answer, if any, is incorrect. Therefore, you need to remember the sequence.

Application of similar

Such elements include numbers with a variable of the same order or the same degree. There are also so-called free terms that do not have a letter designation for the unknown next to them.

The point is that in the absence of parentheses you can simplify the expression by adding or subtracting similar.

A few illustrative examples:

• 8x 2 and 3x 2 - both numbers have the same second order variable, so they are similar and when added they simplify to (8+3)x 2 =11x 2, while when subtracted they get (8-3)x 2 =5x 2 ;
• 4x 3 and 6x - and here “x” has different degrees;
• 2y 7 and 33x 7 - contain different variables, therefore, as in the previous case, they are not similar.

Factoring a number

This little mathematical trick, if you learn to use it correctly, will more than once help you cope with a tricky problem in the future. And it’s not difficult to understand how the “system” works: decomposition is the product of several elements, the calculation of which gives original value . So 20 can be represented as 20x1, 2x10, 5x4, 2x5x2, or some other way.

On a note: Factors are always the same as divisors. So you need to look for a working “pair” for decomposition among the numbers into which the original is divisible without a remainder.

This operation can be performed both with free terms and with numbers in a variable. The main thing is not to lose the latter during calculations - even after decomposition, the unknown cannot just “go nowhere.” It remains at one of the multipliers:

• 15x=3(5x);
• 60y 2 =(15y 2)4.

Prime numbers that can only be divided by themselves or 1 are never expanded - it makes no sense.

Basic methods of simplification

The first thing your eye catches:

• the presence of parentheses;
• fractions;
• roots.

Algebraic examples in school curriculum are often written with the idea that they can be beautifully simplified.

Calculations in parentheses

Pay close attention to the sign in front of the brackets! Multiplication or division is applied to each element inside, and a minus sign reverses the existing “+” or “-” signs.

Brackets are calculated according to the rules or using abbreviated multiplication formulas, after which similar ones are given.

Reducing Fractions

Reduce fractions It's also easy. They themselves “willingly run away” every once in a while, as soon as operations are carried out to bring in such members. But you can simplify the example even before that: pay attention to the numerator and denominator. They often contain explicit or hidden elements that can be mutually reduced. True, if in the first case you just need to cross out the unnecessary, in the second you will have to think, bringing part of the expression to form for simplification. Methods used:

• searching and bracketing the largest common divisor at the numerator and denominator;
• dividing each top element by the denominator.

When an expression or part of it is under the root, the primary task of simplification is almost similar to the case with fractions. It is necessary to look for ways to completely get rid of it or, if this is not possible, to minimize the sign that interferes with calculations. For example, up to the unobtrusive √(3) or √(7).

The right way simplify the radical expression - try to factor it, some of which extend beyond the sign. A good example: √(90)=√(9×10) =√(9)×√(10)=3√(10).

Other little tricks and nuances:

• this simplification operation can be carried out with fractions, taking it out of the sign both as a whole and separately as the numerator or denominator;
• Part of the sum or difference cannot be expanded and taken beyond the root;
• When working with variables, be sure to take into account its degree, it must be equal to or multiple of the root for the possibility of removal: √(x 2 y)=x√(y), √(x 3)=√(x 2 ×x)=x√(x);
• sometimes it is possible to get rid of the radical variable by raising it to a fractional power: √(y 3)=y 3/2.

Simplifying a Power Expression

If in the case of simple calculations by minus or plus the examples are simplified by citing similar ones, then what to do when multiplying or dividing variables with different degrees? They can be easily simplified by remembering two main points:

1. If there is a multiplication sign between the variables, the powers add up.
2. When they are divided by each other, the same power of the denominator is subtracted from the power of the numerator.

The only condition for such simplification is that both terms have the same basis. Examples for clarity:

• 5x 2 ×4x 7 +(y 13 /y 11)=(5×4)x 2+7 +y 13- 11 =20x 9 +y 2;
• 2z 3 +z×z 2 -(3×z 8 /z 5)=2z 3 +z 1+2 -(3×z 8-5)=2z 3 +z 3 -3z 3 =3z 3 -3z 3 = 0.

We note that operations with numeric values ​​in front of variables occur according to the usual mathematical rules. And if you look closely, it becomes clear that the power elements of the expression “work” in a similar way:

• raising a term to a power means multiplying it by itself a certain number of times, i.e. x 2 =x×x;
• division is similar: if you expand the powers of the numerator and denominator, then some of the variables will be canceled, while the remaining ones are “collected,” which is equivalent to subtraction.

As with anything, simplifying algebraic expressions requires not only knowledge of the basics, but also practice. After just a few lessons, examples that once seemed complex will be reduced without much difficulty, turning into short and easily solved ones.

Video

This video will help you understand and remember how expressions are simplified.

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