Note 1
A Boolean function can be written using a Boolean expression and can then be moved to a logic circuit. It is necessary to simplify logical expressions in order to obtain the simplest (and therefore cheaper) logical circuit possible. In fact, a logical function, a logical expression and a logical circuit are three different languages that talk about one entity.
To simplify logical expressions use laws of algebra logic.
Some transformations are similar to transformations of formulas in classical algebra (taking the common factor out of brackets, using commutative and combinational laws, etc.), while other transformations are based on properties that the operations of classical algebra do not have (using the distributive law for conjunction, laws of absorption, gluing, de Morgan's rules, etc.).
The laws of logical algebra are formulated for basic logical operations - “NOT” – inversion (negation), “AND” – conjunction (logical multiplication) and “OR” – disjunction (logical addition).
The law of double negation means that the “NOT” operation is reversible: if you apply it twice, then in the end the logical value will not change.
The law of excluded middle states that any logical expression is either true or false (“there is no third”). Therefore, if $A=1$, then $\bar(A)=0$ (and vice versa), which means that the conjunction of these quantities is always equal to zero, and the disjunction is always equal to one.
$((A + B) → C) \cdot (B → C \cdot D) \cdot C.$
Let's simplify this formula:
Figure 3.
It follows that $A = 0$, $B = 1$, $C = 1$, $D = 1$.
Answer: Students $B$, $C$ and $D$ play chess, but student $A$ does not play.
When simplifying logical expressions, you can perform the following sequence of actions:
- Replace all “non-basic” operations (equivalence, implication, exclusive OR, etc.) with their expressions through the basic operations of inversion, conjunction and disjunction.
- Expand inversions of complex expressions according to De Morgan's rules in such a way that negation operations remain only for individual variables.
- Then simplify the expression using opening brackets, placing common factors outside brackets and other laws of logical algebra.
Example 2
Here, De Morgan's rule, the distributive law, the law of the excluded middle, the commutative law, the law of repetition, again the commutative law and the law of absorption are used successively.
Among the various expressions that are considered in algebra, sums of monomials occupy an important place. Here are examples of such expressions:
\(5a^4 - 2a^3 + 0.3a^2 - 4.6a + 8\)
\(xy^3 - 5x^2y + 9x^3 - 7y^2 + 6x + 5y - 2\)
The sum of monomials is called a polynomial. The terms in a polynomial are called terms of the polynomial. Monomials are also classified as polynomials, considering a monomial to be a polynomial consisting of one member.
For example, a polynomial
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 \)
can be simplified.
Let us represent all terms in the form of monomials of the standard form:
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 = \)
\(= 8b^5 - 14b^5 + 3b^2 -8b -3b^2 + 16\)
Let us present similar terms in the resulting polynomial:
\(8b^5 -14b^5 +3b^2 -8b -3b^2 + 16 = -6b^5 -8b + 16 \)
The result is a polynomial, all terms of which are monomials of the standard form, and among them there are no similar ones. Such polynomials are called polynomials of standard form.
For degree of polynomial of a standard form take the highest of the powers of its members. Thus, the binomial \(12a^2b - 7b\) has the third degree, and the trinomial \(2b^2 -7b + 6\) has the second.
Typically, the terms of standard form polynomials containing one variable are arranged in descending order of exponents. For example:
\(5x - 18x^3 + 1 + x^5 = x^5 - 18x^3 + 5x + 1\)
The sum of several polynomials can be transformed (simplified) into a polynomial of standard form.
Sometimes the terms of a polynomial need to be divided into groups, enclosing each group in parentheses. Since bracketing is the inverse transformation of opening brackets, it is easy to formulate rules for opening brackets:
If a “+” sign is placed before the brackets, then the terms enclosed in brackets are written with the same signs.
If a “-” sign is placed before the brackets, then the terms enclosed in brackets are written with opposite signs.
Transformation (simplification) of the product of a monomial and a polynomial
Using the distributive property of multiplication, you can transform (simplify) the product of a monomial and a polynomial into a polynomial. For example:
\(9a^2b(7a^2 - 5ab - 4b^2) = \)
\(= 9a^2b \cdot 7a^2 + 9a^2b \cdot (-5ab) + 9a^2b \cdot (-4b^2) = \)
\(= 63a^4b - 45a^3b^2 - 36a^2b^3 \)
The product of a monomial and a polynomial is identically equal to the sum of the products of this monomial and each of the terms of the polynomial.
This result is usually formulated as a rule.
To multiply a monomial by a polynomial, you must multiply that monomial by each of the terms of the polynomial.
We have already used this rule several times to multiply by a sum.
Product of polynomials. Transformation (simplification) of the product of two polynomials
In general, the product of two polynomials is identically equal to the sum of the product of each term of one polynomial and each term of the other.
Usually the following rule is used.
To multiply a polynomial by a polynomial, you need to multiply each term of one polynomial by each term of the other and add the resulting products.
Abbreviated multiplication formulas. Sum squares, differences and difference of squares
You have to deal with some expressions in algebraic transformations more often than others. Perhaps the most common expressions are \((a + b)^2, \; (a - b)^2 \) and \(a^2 - b^2 \), i.e. the square of the sum, the square of the difference and difference of squares. You noticed that the names of these expressions seem to be incomplete, for example, \((a + b)^2 \) is, of course, not just the square of the sum, but the square of the sum of a and b. However, the square of the sum of a and b does not occur very often; as a rule, instead of the letters a and b, it contains various, sometimes quite complex, expressions.
The expressions \((a + b)^2, \; (a - b)^2 \) can be easily converted (simplified) into polynomials of the standard form; in fact, you have already encountered this task when multiplying polynomials:
\((a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = \)
\(= a^2 + 2ab + b^2 \)
It is useful to remember the resulting identities and apply them without intermediate calculations. Brief verbal formulations help this.
\((a + b)^2 = a^2 + b^2 + 2ab \) - the square of the sum is equal to the sum of the squares and the double product.
\((a - b)^2 = a^2 + b^2 - 2ab \) - the square of the difference is equal to the sum of squares without the doubled product.
\(a^2 - b^2 = (a - b)(a + b) \) - the difference of squares is equal to the product of the difference and the sum.
These three identities allow one to replace its left-hand parts with right-hand ones in transformations and vice versa - right-hand parts with left-hand ones. The most difficult thing is to see the corresponding expressions and understand how the variables a and b are replaced in them. Let's look at several examples of using abbreviated multiplication formulas.
Entry level
Converting Expressions. Detailed Theory (2019)
Converting Expressions
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 lesson, you need to be able to handle fractions and factor polynomials. Therefore, first, 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.
Basic 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 are terms (monomials) with the same letter part. For example, in sum, similar terms are and.
Do you remember?
To bring similar means to add several similar terms to each other and get 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. Then:
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 needs to be factorized, that is, presented as a product. This is especially important in fractions: in order to be able to reduce a 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, decide a few examples(needs to be factorized):
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.
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 consolidate, solve a few yourself examples:
Answers:
1. I hope you didn’t immediately rush to cut and? It was still not enough to “reduce” units like this:
The first step should be factorization:
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:
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 you multiply 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:
Here we need to remember one more thing - the difference of cubes:
Please note that the denominator of the second fraction does not contain the formula “square of the sum”! The square of the sum would look like this: .
A is the so-called incomplete square of the sum: the second term in it is the product of the first and last, and not their double product. The partial square of the sum is one of the factors in the expansion of the difference of cubes:
What to do if there are already three fractions?
Yes, the same thing! First of all, let’s make sure that the maximum number of factors in the denominators is the same:
Please note: if you change the signs inside one bracket, the sign in front of the fraction changes to the opposite. When we change the signs in the second bracket, the sign in front of the fraction changes again to the opposite. As a result, it (the sign in front of the fraction) has not changed.
We write out the entire first denominator into the common denominator, and then add to it all the factors that have not yet been written, from the second, and then from the third (and so on, if there are more fractions). That is, it turns out like this:
Hmm... It’s clear what to do with fractions. But what about the two?
It's simple: you know how to add fractions, right? So, we need to make two become a fraction! Let's remember: a fraction is a division operation (the numerator is divided by the denominator, in case you forgot). And there is nothing easier than dividing a number by. In this case, the number itself will not change, but will turn into a fraction:
Just what you need!
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 the 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.
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:
Now I’ll show you the process, tinting the current action in red:
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:
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:
;
Using any language, you can express the same information in different words and phrases. Mathematical language is no exception. But the same expression can be equivalently written in different ways. And in some situations, one of the entries is simpler. We'll talk about simplifying expressions in this lesson.
People communicate in different languages. For us, an important comparison is the pair “Russian language - mathematical language”. The same information can be communicated in different languages. But, besides this, it can be pronounced in different ways in one language.
For example: “Petya is friends with Vasya”, “Vasya is friends with Petya”, “Petya and Vasya are friends”. Said differently, but the same thing. From any of these phrases we would understand what we are talking about.
Let's look at this phrase: “The boy Petya and the boy Vasya are friends.” We understand what we are talking about. However, we don't like the sound of this phrase. Can't we simplify it, say the same thing, but simpler? “Boy and boy” - you can say once: “The boys Petya and Vasya are friends.”
“Boys”... Isn’t it clear from their names that they are not girls? We remove the “boys”: “Petya and Vasya are friends.” And the word “friends” can be replaced with “friends”: “Petya and Vasya are friends.” As a result, the first, long, ugly phrase was replaced with an equivalent statement that is easier to say and easier to understand. We have simplified this phrase. To simplify means to say it more simply, but not to lose or distort the meaning.
In mathematical language, roughly the same thing happens. The same thing can be said, written differently. What does it mean to simplify an expression? This means that for the original expression there are many equivalent expressions, that is, those that mean the same thing. And from all this variety we must choose the simplest, in our opinion, or the most suitable for our further purposes.
For example, consider the numeric expression . It will be equivalent to .
It will also be equivalent to the first two: 
.
It turns out that we have simplified our expressions and found the shortest equivalent expression.
For numeric expressions, you always need to do everything and get the equivalent expression as a single number.
Let's look at an example of a literal expression . Obviously, it will be simpler.
When simplifying literal expressions, it is necessary to perform all possible actions.
Is it always necessary to simplify an expression? No, sometimes it will be more convenient for us to have an equivalent but longer entry.
Example: you need to subtract a number from a number.
It is possible to calculate, but if the first number were represented by its equivalent notation: , then the calculations would be instantaneous: .
That is, a simplified expression is not always beneficial for us for further calculations.
Nevertheless, very often we are faced with a task that just sounds like “simplify the expression.”
Simplify the expression: .
Solution
1) Perform the actions in the first and second brackets: .
2) Let's calculate the products: .
Obviously, the last expression has a simpler form than the initial one. We've simplified it.
In order to simplify the expression, it must be replaced with an equivalent (equal).
To determine the equivalent expression you need:
1) perform all possible actions,
2) use the properties of addition, subtraction, multiplication and division to simplify calculations.
Properties of addition and subtraction:
1. Commutative property of addition: rearranging the terms does not change the sum.
2. 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 numbers to the first number.
3. The property of subtracting a sum from a number: to subtract a sum from a number, you can subtract each term separately.
Properties of multiplication and division
1. Commutative property of multiplication: rearranging the factors does not change the product.
2. Combinative property: to multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second factor.
3. Distributive property of multiplication: in order to multiply a number by a sum, you need to multiply it by each term separately.
Let's see how we actually do mental calculations.
Calculate:
Solution
1) Let's imagine how
2) Let's imagine the first factor as a sum of bit terms and perform the multiplication:
3) you can imagine how and perform multiplication:
4) Replace the first factor with an equivalent sum:
The distribution law can also be used in the opposite direction: .
Follow these steps:
1) 
2) 
Solution
1) For convenience, you can use the distributive law, but use it in the opposite direction - take the common factor out of brackets.
2) Let’s take the common factor out of brackets
It is necessary to buy linoleum for the kitchen and hallway. Kitchen area - , hallway - . There are three types of linoleums: for, and rubles for. How much will each of the three types of linoleum cost? (Fig. 1)
Rice. 1. Illustration for the problem statement
Solution
Method 1. You can separately find out how much money it will take to buy linoleum for the kitchen, and then in the hallway and add up the resulting products.
Entry level
Converting Expressions. Detailed Theory (2019)
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!)
Important note!If you see gobbledygook instead of formulas, clear your cache. To do this, press CTRL+F5 (on Windows) or Cmd+R (on Mac).
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:
1. I hope you didn’t immediately rush to cut and? It was still not enough to “reduce” units like this:
The first step should be factorization:
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 you multiply 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:
Here we need to remember one more thing - the difference of cubes:
Please note that the denominator of the second fraction does not contain the formula “square of the sum”! The square of the sum would look like this: .
A is the so-called incomplete square of the sum: the second term in it is the product of the first and last, and not their double product. The partial square of the sum is one of the factors in the expansion of the difference of cubes:
What to do if there are already three fractions?
Yes, the same thing! First of all, let’s make sure that the maximum number of factors in the denominators is the same:
Please note: if you change the signs inside one bracket, the sign in front of the fraction changes to the opposite. When we change the signs in the second bracket, the sign in front of the fraction changes again to the opposite. As a result, it (the sign in front of the fraction) has not changed.
We write out the entire first denominator into the common denominator, and then add to it all the factors that have not yet been written, from the second, and then from the third (and so on, if there are more fractions). That is, it turns out like this:
Hmm... It’s clear what to do with fractions. But what about the two?
It's simple: you know how to add fractions, right? So, we need to make two become a fraction! Let's remember: a fraction is a division operation (the numerator is divided by the denominator, in case you forgot). And there is nothing easier than dividing a number by. In this case, the number itself will not change, but will turn into a fraction:
Just what you need!
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 the 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:
Now I’ll show you the process, tinting the current action in red:
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:
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