Simplify the equation. Simplifying Boolean Expressions
Among the various expressions that are considered in algebra are important place occupy sums of monomials. 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's represent all terms in the form of monomials standard view:
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 = \)
\(= 8b^5 - 14b^5 + 3b^2 -8b -3b^2 + 16\)
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.
\((a + b)^2 = a^2 + b^2 + 2ab \) - square of the sum equal to the sum squares and double the 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.
A literal expression (or variable expression) is a mathematical expression that consists of numbers, letters, and mathematical symbols. For example, the following expression is literal:
a+b+4
Using alphabetic expressions you can write laws, formulas, equations and functions. The ability to manipulate letter expressions is the key to good knowledge of algebra and higher mathematics.
Any serious problem in mathematics comes down to solving equations. And in order to be able to solve equations, you need to be able to work with literal expressions.
To work with literal expressions, you need to be well-versed in basic arithmetic: addition, subtraction, multiplication, division, basic laws of mathematics, fractions, operations with fractions, proportions. And not just study, but understand thoroughly.
Lesson contentVariables
Letters that are contained in literal expressions are called variables. For example, in the expression a+b+4 the variables are the letters a And b. If we substitute any numbers instead of these variables, then the literal expression a+b+4 will turn into a numerical expression whose value can be found.
Numbers that are substituted for variables are called values of variables. For example, let's change the values of the variables a And b. The equal sign is used to change values
a = 2, b = 3
We have changed the values of the variables a And b. Variable a assigned a value 2 , variable b assigned a value 3 . As a result, the literal expression a+b+4 turns into a regular numeric expression 2+3+4 whose value can be found:
2 + 3 + 4 = 9
When variables are multiplied, they are written together. For example, record ab means the same as the entry a×b. If we substitute the variables a And b numbers 2 And 3 , then we get 6
2 × 3 = 6
You can also write together the multiplication of a number by an expression in parentheses. For example, instead of a×(b + c) can be written down a(b + c). Applying the distribution law of multiplication, we obtain a(b + c)=ab+ac.
Odds
In literal expressions you can often find a notation in which a number and a variable are written together, for example 3a. This is actually a shorthand for multiplying the number 3 by a variable. a and this entry looks like 3×a .
In other words, the expression 3a is the product of the number 3 and the variable a. Number 3 in this work they call coefficient. This coefficient shows how many times the variable will be increased a. This expression can be read as " a three times" or "three times A", or "increase the value of a variable a three times", but most often read as "three a«
For example, if the variable a equal to 5 , then the value of the expression 3a will be equal to 15.
3 × 5 = 15
Speaking in simple language, the coefficient is the number that comes before the letter (before the variable).
There can be several letters, for example 5abc. Here the coefficient is the number 5 . This coefficient shows that the product of variables abc increases fivefold. This expression can be read as " abc five times" or "increase the value of the expression abc five times" or "five abc«.
If instead of variables abc substitute the numbers 2, 3 and 4, then the value of the expression 5abc will be equal 120
5 × 2 × 3 × 4 = 120
You can mentally imagine how the numbers 2, 3 and 4 were first multiplied, and the resulting value increased fivefold:
The sign of the coefficient refers only to the coefficient and does not apply to the variables.
Consider the expression −6b. Minus before the coefficient 6 , applies only to the coefficient 6 , and does not belong to the variable b. Understanding this fact will allow you not to make mistakes in the future with signs.
Let's find the value of the expression −6b at b = 3.
−6b −6×b. For clarity, let us write the expression −6b in expanded form and substitute the value of the variable b
−6b = −6 × b = −6 × 3 = −18
Example 2. Find the value of an expression −6b at b = −5
Let's write down the expression −6b in expanded form
−6b = −6 × b = −6 × (−5) = 30
Example 3. Find the value of an expression −5a+b at a = 3 And b = 2
−5a+b this is a short form for −5 × a + b, so for clarity we write the expression −5×a+b in expanded form and substitute the values of the variables a And b
−5a + b = −5 × a + b = −5 × 3 + 2 = −15 + 2 = −13
Sometimes letters are written without a coefficient, for example a or ab. In this case, the coefficient is unity:
but traditionally the unit is not written down, so they simply write a or ab
If there is a minus before the letter, then the coefficient is a number −1 . For example, the expression −a actually looks like −1a. This is the product of minus one and the variable a. It turned out like this:
−1 × a = −1a
There is a small catch here. In expression −a minus sign in front of the variable a actually refers to an "invisible unit" rather than a variable a. Therefore, you should be careful when solving problems.
For example, if given the expression −a and we are asked to find its value at a = 2, then at school we substituted a two instead of a variable a and received an answer −2 , without focusing too much on how it turned out. In fact, minus one was multiplied by positive number 2
−a = −1 × a
−1 × a = −1 × 2 = −2
If given the expression −a and you need to find its value at a = −2, then we substitute −2 instead of a variable a
−a = −1 × a
−1 × a = −1 × (−2) = 2
To avoid mistakes, at first invisible units can be written down explicitly.
Example 4. Find the value of an expression abc at a=2 , b=3 And c=4
Expression abc 1×a×b×c. For clarity, let us write the expression abc a, b And c
1 × a × b × c = 1 × 2 × 3 × 4 = 24
Example 5. Find the value of an expression abc at a=−2 , b=−3 And c=−4
Let's write down the expression abc in expanded form and substitute the values of the variables a, b And c
1 × a × b × c = 1 × (−2) × (−3) × (−4) = −24
Example 6. Find the value of an expression − abc at a=3 , b=5 and c=7
Expression − abc this is a short form for −1×a×b×c. For clarity, let us write the expression − abc in expanded form and substitute the values of the variables a, b And c
−abc = −1 × a × b × c = −1 × 3 × 5 × 7 = −105
Example 7. Find the value of an expression − abc at a=−2 , b=−4 and c=−3
Let's write down the expression − abc in expanded form:
−abc = −1 × a × b × c
Let's substitute the values of the variables a , b And c
−abc = −1 × a × b × c = −1 × (−2) × (−4) × (−3) = 24
How to determine the coefficient
Sometimes you need to solve a problem in which you need to determine the coefficient of an expression. In principle, this task is very simple. It is enough to be able to multiply numbers correctly.
To determine the coefficient in an expression, you need to separately multiply the numbers included in this expression and separately multiply the letters. The resulting numerical factor will be the coefficient.
Example 1. 7m×5a×(−3)×n
The expression consists of several factors. This can be clearly seen if you write the expression in expanded form. That is, the works 7m And 5a write it in the form 7×m And 5×a
7 × m × 5 × a × (−3) × n
Let's apply the associative law of multiplication, which allows you to multiply factors in any order. Namely, we will separately multiply the numbers and separately multiply the letters (variables):
−3 × 7 × 5 × m × a × n = −105man
The coefficient is −105 . After completion, it is advisable to arrange the letter part in alphabetical order:
−105amn
Example 2. Determine the coefficient in the expression: −a×(−3)×2
−a × (−3) × 2 = −3 × 2 × (−a) = −6 × (−a) = 6a
The coefficient is 6.
Example 3. Determine the coefficient in the expression: 
Let's multiply numbers and letters separately:
The coefficient is −1. Please note that the unit is not written down, since it is customary not to write the coefficient 1.
These seemingly simplest tasks can play a very cruel joke on us. It often turns out that the sign of the coefficient is set incorrectly: either the minus is missing or, on the contrary, it was set in vain. To avoid these annoying mistakes, it must be studied at a good level.
Addends in literal expressions
When adding several numbers, the sum of these numbers is obtained. Numbers that add are called addends. There can be several terms, for example:
1 + 2 + 3 + 4 + 5
When an expression consists of terms, it is much easier to evaluate because adding is easier than subtracting. But the expression can contain not only addition, but also subtraction, for example:
1 + 2 − 3 + 4 − 5
In this expression, the numbers 3 and 5 are subtrahends, not addends. But nothing prevents us from replacing subtraction with addition. Then we again get an expression consisting of terms:
1 + 2 + (−3) + 4 + (−5)
It doesn’t matter that the numbers −3 and −5 now have a minus sign. The main thing is that all the numbers in this expression are connected by an addition sign, that is, the expression is a sum.
Both expressions 1 + 2 − 3 + 4 − 5 And 1 + 2 + (−3) + 4 + (−5) equal to the same value - minus one
1 + 2 − 3 + 4 − 5 = −1
1 + 2 + (−3) + 4 + (−5) = −1
Thus, the meaning of the expression will not suffer if we replace subtraction with addition somewhere.
You can also replace subtraction with addition in literal expressions. For example, consider the following expression:
7a + 6b − 3c + 2d − 4s
7a + 6b + (−3c) + 2d + (−4s)
For any values of variables a, b, c, d And s expressions 7a + 6b − 3c + 2d − 4s And 7a + 6b + (−3c) + 2d + (−4s) will be equal to the same value.
You must be prepared for the fact that a teacher at school or a teacher at an institute may call even numbers (or variables) that are not addends.
For example, if the difference is written on the board a − b, then the teacher will not say that a is a minuend, and b- subtractable. He will call both variables with one common word - terms. And all because the expression of the form a − b the mathematician sees how the sum a+(−b). In this case, the expression becomes a sum, and the variables a And (−b) become terms.
Similar terms
Similar terms- these are terms that have the same letter part. For example, consider the expression 7a + 6b + 2a. Components 7a And 2a have the same letter part - variable a. So the terms 7a And 2a are similar.
Typically, similar terms are added to simplify an expression or solve an equation. This operation is called bringing similar terms.
To bring similar terms, you need to add the coefficients of these terms, and multiply the resulting result by the common letter part.
For example, let us present similar terms in the expression 3a + 4a + 5a. In this case, all terms are similar. Let's add up their coefficients and multiply the result by the common letter part - by the variable a
3a + 4a + 5a = (3 + 4 + 5)×a = 12a
Similar terms are usually brought up in mind and the result is written down immediately:
3a + 4a + 5a = 12a
Also, one can reason as follows:
There were 3 variables a , 4 more variables a and 5 more variables a were added to them. As a result, we got 12 variables a
Let's look at several examples of bringing similar terms. Considering that this topic is very important, at first we will write down every little detail in detail. Even though everything is very simple here, most people make many mistakes. Mostly due to inattention, not ignorance.
Example 1. 3a + 2a + 6a + 8 a
Let's add up the coefficients in this expression and multiply the resulting result by the common letter part:
3a + 2a + 6a + 8a = (3 + 2 + 6 + 8) × a = 19a
design (3 + 2 + 6 + 8)×a You don’t have to write it down, so we’ll write down the answer right away
3a + 2a + 6a + 8a = 19a
Example 2. Give similar terms in the expression 2a+a
Second term a written without a coefficient, but in fact there is a coefficient in front of it 1 , which we do not see because it is not recorded. So the expression looks like this:
2a + 1a
Now let's present similar terms. That is, we add up the coefficients and multiply the result by the common letter part:
2a + 1a = (2 + 1) × a = 3a
Let's write down the solution briefly:
2a + a = 3a
2a+a, you can think differently:
Example 3. Give similar terms in the expression 2a−a
Let's replace subtraction with addition:
2a + (−a)
Second term (−a) written without a coefficient, but in reality it looks like (−1a). Coefficient −1 again invisible due to the fact that it is not recorded. So the expression looks like this:
2a + (−1a)
Now let's present similar terms. Let's add the coefficients and multiply the result by the common letter part:
2a + (−1a) = (2 + (−1)) × a = 1a = a
Usually written shorter:
2a − a = a
Giving similar terms in the expression 2a−a You can think differently:
There were 2 variables a, subtract one variable a, in the end there was only one variable a left
Example 4. Give similar terms in the expression 6a − 3a + 4a − 8a
6a − 3a + 4a − 8a = 6a + (−3a) + 4a + (−8a)
Now let's present similar terms. Let's add the coefficients and multiply the result by the total letter part
(6 + (−3) + 4 + (−8)) × a = −1a = −a
Let's write down the solution briefly:
6a − 3a + 4a − 8a = −a
There are expressions that contain several different groups of similar terms. For example, 3a + 3b + 7a + 2b. For such expressions, the same rules apply as for the others, namely, adding the coefficients and multiplying the result by the common letter part. But to avoid mistakes, it’s convenient different groups The terms are highlighted with different lines.
For example, in the expression 3a + 3b + 7a + 2b those terms that contain a variable a, can be underlined with one line, and those terms that contain a variable b, can be emphasized with two lines:
Now we can present similar terms. That is, add the coefficients and multiply the resulting result by the total letter part. This must be done for both groups of terms: for terms containing a variable a and for terms containing a variable b.
3a + 3b + 7a + 2b = (3+7)×a + (3 + 2)×b = 10a + 5b
Again, we repeat, the expression is simple, and similar terms can be given in mind:
3a + 3b + 7a + 2b = 10a + 5b
Example 5. Give similar terms in the expression 5a − 6a −7b + b
Let's replace subtraction with addition where possible:
5a − 6a −7b + b = 5a + (−6a) + (−7b) + b
Let us underline similar terms with different lines. Terms containing variables a we underline with one line, and the terms are the contents of the variables b, underline with two lines:
Now we can present similar terms. That is, add the coefficients and multiply the resulting result by the common letter part:
5a + (−6a) + (−7b) + b = (5 + (−6))×a + ((−7) + 1)×b = −a + (−6b)
If the expression contains ordinary numbers without letter factors, then they are added separately.
Example 6. Give similar terms in the expression 4a + 3a − 5 + 2b + 7
Let's replace subtraction with addition where possible:
4a + 3a − 5 + 2b + 7 = 4a + 3a + (−5) + 2b + 7
Let us present similar terms. Numbers −5 And 7 do not have letter factors, but they are similar terms - they just need to be added. And the term 2b will remain unchanged, since it is the only one in this expression that has a letter factor b, and there is nothing to add it with:
4a + 3a + (−5) + 2b + 7 = (4 + 3)×a + 2b + (−5) + 7 = 7a + 2b + 2
Let's write down the solution briefly:
4a + 3a − 5 + 2b + 7 = 7a + 2b + 2
The terms can be ordered so that those terms that have the same letter part are located in the same part of the expression.
Example 7. Give similar terms in the expression 5t+2x+3x+5t+x
Since the expression is a sum of several terms, this allows us to evaluate it in any order. Therefore, the terms containing the variable t, can be written at the beginning of the expression, and the terms containing the variable x at the end of the expression:
5t + 5t + 2x + 3x + x
Now we can present similar terms:
5t + 5t + 2x + 3x + x = (5+5)×t + (2+3+1)×x = 10t + 6x
Let's write down the solution briefly:
5t + 2x + 3x + 5t + x = 10t + 6x
The sum of opposite numbers is zero. This rule also works for literal expressions. If the expression contains identical terms, but with opposite signs, then you can get rid of them at the stage of reducing similar terms. In other words, simply eliminate them from the expression, since their sum is zero.
Example 8. Give similar terms in the expression 3t − 4t − 3t + 2t
Let's replace subtraction with addition where possible:
3t − 4t − 3t + 2t = 3t + (−4t) + (−3t) + 2t
Components 3t And (−3t) are opposite. The sum of opposite terms is zero. If we remove this zero from the expression, the value of the expression will not change, so we will remove it. And we will remove it by simply crossing out the terms 3t And (−3t)
As a result, we will be left with the expression (−4t) + 2t. In this expression, you can add similar terms and get the final answer:
(−4t) + 2t = ((−4) + 2)×t = −2t
Let's write down the solution briefly:
Simplifying Expressions
"simplify the expression" and below is the expression that needs to be simplified. Simplify an expression means making it simpler and shorter.
In fact, we've already been simplifying expressions when we've reduced fractions. After reduction, the fraction became shorter and easier to understand.
Consider the following example. Simplify the expression.
This task can literally be understood as follows: “Apply any valid actions to this expression, but make it simpler.” .
In this case, you can reduce the fraction, namely, divide the numerator and denominator of the fraction by 2:
What else can you do? You can calculate the resulting fraction. Then we get the decimal fraction 0.5
As a result, the fraction was simplified to 0.5.
The first question you need to ask yourself when solving such problems should be “What can be done?” . Because there are actions that you can do, and there are actions that you cannot do.
Another important point The thing to remember is that the value of the expression should not change after simplifying the expression. Let's return to the expression. This expression represents a division that can be performed. Having performed this division, we get the value of this expression, which is equal to 0.5
But we simplified the expression and got a new simplified expression. The value of the new simplified expression is still 0.5
But we also tried to simplify the expression by calculating it. As a result, we received a final answer of 0.5.
Thus, no matter how we simplify the expression, the value of the resulting expressions is still equal to 0.5. This means that the simplification was carried out correctly at every stage. This is exactly what we should strive for when simplifying expressions - the meaning of the expression should not suffer from our actions.
It is often necessary to simplify literal expressions. The same simplification rules apply to them as for numerical expressions. You can perform any valid actions, as long as the value of the expression does not change.
Let's look at a few examples.
Example 1. Simplify an expression 5.21s × t × 2.5
To simplify this expression, you can multiply the numbers separately and multiply the letters separately. This task is very similar to the one we looked at when we learned to determine the coefficient:
5.21s × t × 2.5 = 5.21 × 2.5 × s × t = 13.025 × st = 13.025st
So the expression 5.21s × t × 2.5 simplified to 13,025st.
Example 2. Simplify an expression −0.4 × (−6.3b) × 2
Second piece (−6.3b) can be translated into a form understandable to us, namely written in the form ( −6,3)×b , then multiply the numbers separately and multiply the letters separately:
− 0,4 × (−6.3b) × 2 = − 0,4 × (−6.3) × b × 2 = 5.04b
So the expression −0.4 × (−6.3b) × 2 simplified to 5.04b
Example 3. Simplify an expression 
Let's write this expression in more detail to clearly see where the numbers are and where the letters are:
Now let’s multiply the numbers separately and multiply the letters separately:
So the expression simplified to −abc. This solution can be written briefly:
When simplifying expressions, fractions can be reduced during the solution process, and not at the very end, as we did with ordinary fractions. For example, if in the course of solving we come across an expression of the form , then it is not at all necessary to calculate the numerator and denominator and do something like this:
A fraction can be reduced by choosing a factor in both the numerator and the denominator and reducing these factors by their greatest common factor. In other words, use in which we do not describe in detail what the numerator and denominator were divided into.
For example, in the numerator the factor is 12 and in the denominator the factor 4 can be reduced by 4. We keep the four in our mind, and dividing 12 and 4 by this four, we write down the answers next to these numbers, having first crossed them out
Now you can multiply the resulting small factors. In this case, there are few of them and you can multiply them in your mind:
Over time, you may find that when solving a particular problem, expressions begin to “get fat,” so it is advisable to get used to quick calculations. What can be calculated in the mind must be calculated in the mind. What can be quickly reduced must be reduced quickly.
Example 4. Simplify an expression 
So the expression simplified to
Example 5. Simplify an expression 
Let's multiply the numbers separately and the letters separately:
So the expression simplified to mn.
Example 6. Simplify an expression 
Let's write this expression in more detail to clearly see where the numbers are and where the letters are:
Now let’s multiply the numbers separately and the letters separately. For ease of calculation, the decimal fraction −6.4 and mixed number can be converted to ordinary fractions:
So the expression simplified to
The solution for this example can be written much shorter. It will look like this:
Example 7. Simplify an expression 
Let's multiply numbers separately and letters separately. For ease of calculation, a mixed number and decimals 0.1 and 0.6 can be converted to ordinary fractions:
So the expression simplified to abcd. If you skip the details, this solution can be written much shorter:
Notice how the fraction has been reduced. New factors that are obtained as a result of reduction of previous factors can also be reduced.
Now let's talk about what not to do. When simplifying expressions, it is strictly forbidden to multiply numbers and letters if the expression is a sum and not a product.
For example, if you want to simplify the expression 5a+4b, then you cannot write it like this:
This is the same as if we were asked to add two numbers and we multiplied them instead of adding them.
When substituting any variable values a And b expression 5a +4b turns into an ordinary numerical expression. Let's assume that the variables a And b have the following meanings:
a = 2, b = 3
Then the value of the expression will be equal to 22
5a + 4b = 5 × 2 + 4 × 3 = 10 + 12 = 22
First, multiplication is performed, and then the results are added. And if we tried to simplify this expression by multiplying numbers and letters, we would get the following:
5a + 4b = 5 × 4 × a × b = 20ab
20ab = 20 × 2 × 3 = 120
It turns out a completely different meaning of the expression. In the first case it worked 22 , in the second case 120 . This means that simplifying the expression 5a+4b was performed incorrectly.
After simplifying the expression, its value should not change with the same values of the variables. If, when substituting any variable values into the original expression, one value is obtained, then after simplifying the expression, the same value should be obtained as before the simplification.
With expression 5a+4b there's really nothing you can do. It doesn't simplify it.
If an expression contains similar terms, then they can be added if our goal is to simplify the expression.
Example 8. Simplify an expression 0.3a−0.4a+a
0.3a − 0.4a + a = 0.3a + (−0.4a) + a = (0.3 + (−0.4) + 1)×a = 0.9a
or shorter: 0.3a − 0.4a + a = 0.9a
So the expression 0.3a−0.4a+a simplified to 0.9a
Example 9. Simplify an expression −7.5a − 2.5b + 4a
To simplify this expression, we can add similar terms:
−7.5a − 2.5b + 4a = −7.5a + (−2.5b) + 4a = ((−7.5) + 4)×a + (−2.5b) = −3.5a + (−2.5b)
or shorter −7.5a − 2.5b + 4a = −3.5a + (−2.5b)
Term (−2.5b) remained unchanged because there was nothing to put it with.
Example 10. Simplify an expression 
To simplify this expression, we can add similar terms:
The coefficient was for ease of calculation.
So the expression simplified to
Example 11. Simplify an expression 
To simplify this expression, we can add similar terms:
So the expression simplified to .
IN in this example It would be more appropriate to add the first and last coefficients first. In this case we would have a short solution. It would look like this:
Example 12. Simplify an expression 
To simplify this expression, we can add similar terms:
So the expression simplified to 
.
The term remained unchanged, since there was nothing to add it to.
This solution can be written much shorter. It will look like this:
The short solution skipped the steps of replacing subtraction with addition and detailing how fractions were reduced to a common denominator.
Another difference is that in detailed solution the answer looks like , but in short as . In fact, they are the same expression. The difference is that in the first case, subtraction is replaced by addition, because at the beginning, when we wrote down the solution in detailed form, we replaced subtraction with addition wherever possible, and this replacement was preserved for the answer.
Identities. Identically equal expressions
Once we have simplified any expression, it becomes simpler and shorter. To check whether the simplified expression is correct, it is enough to substitute any variable values first into the previous expression that needed to be simplified, and then into the new one that was simplified. If the value in both expressions is the same, then the simplified expression is true.
Let's consider simplest example. Let it be necessary to simplify the expression 2a×7b. To simplify this expression, you can multiply numbers and letters separately:
2a × 7b = 2 × 7 × a × b = 14ab
Let's check whether we simplified the expression correctly. To do this, let’s substitute any values of the variables a And b first into the first expression that needed to be simplified, and then into the second, which was simplified.
Let the values of the variables a , b will be as follows:
a = 4, b = 5
Let's substitute them into the first expression 2a×7b
Now let’s substitute the same variable values into the expression that resulted from simplification 2a×7b, namely in the expression 14ab
14ab = 14 × 4 × 5 = 280
We see that when a=4 And b=5 value of the first expression 2a×7b and the meaning of the second expression 14ab equal
2a × 7b = 2 × 4 × 7 × 5 = 280
14ab = 14 × 4 × 5 = 280
The same will happen for any other values. For example, let a=1 And b=2
2a × 7b = 2 × 1 × 7 × 2 =28
14ab = 14 × 1 × 2 =28
Thus, for any values expression variables 2a×7b And 14ab are equal to the same value. Such expressions are called identically equal.
We conclude that between the expressions 2a×7b And 14ab you can put an equal sign because they are equal to the same value.
2a × 7b = 14ab
An equality is any expression that is connected by an equal sign (=).
And equality of the form 2a×7b = 14ab called identity.
An identity is an equality that is true for any values of the variables.
Other examples of identities:
a + b = b + a
a(b+c) = ab + ac
a(bc) = (ab)c
Yes, the laws of mathematics that we studied are identities.
True numerical equalities are also identities. For example:
2 + 2 = 4
3 + 3 = 5 + 1
10 = 7 + 2 + 1
When solving a complex problem, in order to make the calculation easier, the complex expression is replaced with a simpler expression that is identically equal to the previous one. This replacement is called identical transformation of the expression or just transforming the expression.
For example, we simplified the expression 2a×7b, and got a simpler expression 14ab. This simplification can be called the identity transformation.
You can often find a task that says "prove that equality is an identity" and then the equality that needs to be proven is given. Usually this equality consists of two parts: the left and right parts of the equality. Our task is to perform identity transformations with one of the parts of the equality and obtain the other part. Or perform identical transformations with both sides of the equality and make sure that both sides of the equality contain the same expressions.
For example, let us prove that the equality 0.5a × 5b = 2.5ab is an identity.
Let's simplify the left side of this equality. To do this, multiply the numbers and letters separately:
0.5 × 5 × a × b = 2.5ab
2.5ab = 2.5ab
As a result of a small identity transformation, the left side of the equality became equal to the right side of the equality. So we have proven that the equality 0.5a × 5b = 2.5ab is an identity.
From identical transformations we learned to add, subtract, multiply and divide numbers, reduce fractions, add similar terms, and also simplify some expressions.
But these are not all identical transformations that exist in mathematics. There are many more identical transformations. We will see this more than once in the future.
Tasks for independent solution:
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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. Essentially, a logical function, a logical expression and a logical circuit are three different languages, telling 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.
Any language can express the same information in different words and revolutions. 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 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: “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 perform all the steps 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. Matching property addition: 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 multiplication: to multiply a number by a sum, you need to multiply it by each addend 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 distributive law can also be used in reverse side: .
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 cost? three types linoleum? (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 put it in the hallway and add up the resulting products.
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.