Rational expressions calculator with letters. Basic methods of simplification

§ 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.

9x + 4 x = (9 + 4)x = 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 in this case are the 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:

A literal expression (or an expression with variables) is mathematical expression, which consists of numbers, letters and symbols of mathematical operations. 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 content

Variables

Letters that are contained in literal expressions are called variables. For example, in the expression a+b+ 4 variables are 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:

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

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 short form entries from −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's a little 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 is set in vain. To avoid these annoying mistakes, 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 a variables, 4 more a variables and 5 more a variables 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. Despite the fact that 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 total 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 various groups 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.

Let's consider next 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 selecting a factor in the numerator and denominator and reducing these factors by their largest common divisor. 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, mixed numbers and decimal fractions 0.1 and 0.6 can be converted to ordinary fractions:

So the expression simplified to abcd. If you skip the details, then this decision 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 are also allowed to 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 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 the 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, since at the beginning when we wrote the solution in in detail, 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 ​​of the 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.

Faithful numerical equalities are also identities. For example:

2 + 2 = 4

3 + 3 = 5 + 1

10 = 7 + 2 + 1

Deciding difficult task 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, left side 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. Identity transformations much more. We will see this more than once in the future.

Tasks for independent solution:

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Mathematical-Calculator-Online v.1.0

The calculator performs following operations: addition, subtraction, multiplication, division, working with decimals, root extraction, exponentiation, percent calculation and other operations.

Solution:

How to use a math calculator

Key	Designation	Explanation
5	numbers 0-9	Arabic numerals. Entering natural integers, zero. To get a negative integer, you must press the +/- key
.	period (comma)	Separator to indicate a decimal fraction. If there is no number before the point (comma), the calculator will automatically substitute a zero before the point. For example: .5 - 0.5 will be written
+	plus sign	Adding numbers (integers, decimals)
-	minus sign	Subtracting numbers (integers, decimals)
÷	division sign	Dividing numbers (integers, decimals)
X	multiplication sign	Multiplying numbers (integers, decimals)
√	root	Extracting the root of a number. When you press the “root” button again, the root is calculated from the result. For example: root of 16 = 4; root of 4 = 2
x 2	squaring	Squaring a number. When you press the "squaring" button again, the result is squared. For example: square 2 = 4; square 4 = 16
1/x	fraction	Output in decimal fractions. The numerator is 1, the denominator is the entered number
%	percent	Getting a percentage of a number. To work, you need to enter: the number from which the percentage will be calculated, the sign (plus, minus, divide, multiply), how many percent in numerical form, the "%" button
(	open parenthesis	An open parenthesis to specify the calculation priority. A closed parenthesis is required. Example: (2+3)*2=10
)	closed parenthesis	A closed parenthesis to specify the calculation priority. Availability required open bracket
±	plus minus	Reverses sign
=	equals	Displays the result of the solution. Also above the calculator, in the “Solution” field, intermediate calculations and the result are displayed.
←	deleting a character	Removes the last character
WITH	reset	Reset button. Completely resets the calculator to position "0"

Algorithm of the online calculator using examples

Addition.

Addition of integers natural numbers { 5 + 7 = 12 }

Addition of integer natural and negative numbers ( 5 + (-2) = 3 )

Adding decimals fractional numbers { 0,3 + 5,2 = 5,5 }

Subtraction.

Subtracting natural integers ( 7 - 5 = 2 )

Subtracting natural and negative integers ( 5 - (-2) = 7 )

Subtracting decimal fractions ( 6.5 - 1.2 = 4.3 )

Multiplication.

Product of natural integers (3 * 7 = 21)

Product of natural and negative integers ( 5 * (-3) = -15 )

Product of decimal fractions ( 0.5 * 0.6 = 0.3 )

Division.

Division of natural integers (27 / 3 = 9)

Division of natural and negative integers (15 / (-3) = -5)

Division of decimal fractions (6.2 / 2 = 3.1)

Extracting the root of a number.

Extracting the root of an integer ( root(9) = 3)

Extracting the root from decimals( root(2.5) = 1.58 )

Extracting the root of a sum of numbers ( root(56 + 25) = 9)

Extracting the root of the difference between numbers (root (32 – 7) = 5)

Squaring a number.

Squaring an integer ( (3) 2 = 9 )

Squaring decimals ((2,2)2 = 4.84)

Conversion to decimal fractions.

Calculating percentages of a number

Increase the number 230 by 15% ( 230 + 230 * 0.15 = 264.5 )

Reduce the number 510 by 35% ( 510 – 510 * 0.35 = 331.5 )

18% of the number 140 is (140 * 0.18 = 25.2)

The power is used to simplify the operation of multiplying a number by itself. For example, instead of writing, you can write 4 5 (\displaystyle 4^(5))(an explanation for this transition is given in the first section of this article). Degrees make it easier to write long or complex expressions or equations; powers are also easily added and subtracted, resulting in a simplified expression or equation (for example, 4 2 ∗ 4 3 = 4 5 (\displaystyle 4^(2)*4^(3)=4^(5))).

Note: if you need to decide exponential equation(in such an equation the unknown is in the exponent), read.

Steps

Solving simple problems with degrees

Multiply the base of the exponent by itself a number of times equal to the exponent. If you need to solve a power problem by hand, rewrite the power as a multiplication operation, where the base of the power is multiplied by itself. For example, given a degree 3 4 (\displaystyle 3^(4)). In this case, the base of power 3 must be multiplied by itself 4 times: 3 ∗ 3 ∗ 3 ∗ 3 (\displaystyle 3*3*3*3). Here are other examples:

First, multiply the first two numbers. For example, 4 5 (\displaystyle 4^(5)) = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4*4*4*4*4). Don't worry - the calculation process is not as complicated as it seems at first glance. First multiply the first two fours and then replace them with the result. Like this:

• 4 5 = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4^(5)=4*4*4*4*4)
  • 4 ∗ 4 = 16 (\displaystyle 4*4=16)

1. Multiply the result (16 in our example) by the next number. Each subsequent result will increase proportionally. In our example, multiply 16 by 4. Like this:

   • 4 5 = 16 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4^(5)=16*4*4*4)
     • 16 ∗ 4 = 64 (\displaystyle 16*4=64)
   • 4 5 = 64 ∗ 4 ∗ 4 (\displaystyle 4^(5)=64*4*4)
     • 64 ∗ 4 = 256 (\displaystyle 64*4=256)
   • 4 5 = 256 ∗ 4 (\displaystyle 4^(5)=256*4)
     • 256 ∗ 4 = 1024 (\displaystyle 256*4=1024)
   • Continue multiplying the result of the first two numbers by the next number until you get your final answer. To do this, multiply the first two numbers, and then multiply the resulting result by the next number in the sequence. This method is valid for any degree. In our example you should get: 4 5 = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 = 1024 (\displaystyle 4^(5)=4*4*4*4*4=1024) .

2. Solve the following problems. Check your answer using a calculator.

   • 8 2 (\displaystyle 8^(2))
   • 3 4 (\displaystyle 3^(4))
   • 10 7 (\displaystyle 10^(7))

3. On your calculator, look for the key labeled "exp" or " x n (\displaystyle x^(n))", or "^". Using this key you will raise a number to a power. It is almost impossible to calculate a degree with a large indicator manually (for example, the degree 9 15 (\displaystyle 9^(15))), but the calculator can easily cope with this task. In Windows 7, the standard calculator can be switched to engineering mode; To do this, click “View” -> “Engineering”. To switch to normal mode, click “View” -> “Normal”.

   • Check the received answer using a search engine (Google or Yandex). Using the "^" key on your computer keyboard, enter the expression into the search engine, which will instantly display the correct answer (and possibly suggest similar expressions for you to study).

   Addition, subtraction, multiplication of powers

   1. You can add and subtract degrees only if they have the same bases. If you need to add powers with the same bases and exponents, then you can replace the addition operation with the multiplication operation. For example, given the expression 4 5 + 4 5 (\displaystyle 4^(5)+4^(5)). Remember that the degree 4 5 (\displaystyle 4^(5)) can be represented in the form 1 ∗ 4 5 (\displaystyle 1*4^(5)); Thus, 4 5 + 4 5 = 1 ∗ 4 5 + 1 ∗ 4 5 = 2 ∗ 4 5 (\displaystyle 4^(5)+4^(5)=1*4^(5)+1*4^(5) =2*4^(5))(where 1 +1 =2). That is, count the number of similar degrees, and then multiply that degree and this number. In our example, raise 4 to the fifth power, and then multiply the resulting result by 2. Remember that the addition operation can be replaced by the multiplication operation, for example, 3 + 3 = 2 ∗ 3 (\displaystyle 3+3=2*3). Here are other examples:

      • 3 2 + 3 2 = 2 ∗ 3 2 (\displaystyle 3^(2)+3^(2)=2*3^(2))
      • 4 5 + 4 5 + 4 5 = 3 ∗ 4 5 (\displaystyle 4^(5)+4^(5)+4^(5)=3*4^(5))
      • 4 5 − 4 5 + 2 = 2 (\displaystyle 4^(5)-4^(5)+2=2)
      • 4 x 2 − 2 x 2 = 2 x 2 (\displaystyle 4x^(2)-2x^(2)=2x^(2))

   2. When multiplying powers with the same basis their indicators are added up (the base does not change). For example, given the expression x 2 ∗ x 5 (\displaystyle x^(2)*x^(5)). In this case, you just need to add the indicators, leaving the base unchanged. Thus, x 2 ∗ x 5 = x 7 (\displaystyle x^(2)*x^(5)=x^(7)). Here is a visual explanation of this rule:

      When raising a power to a power, the exponents are multiplied. For example, a degree is given. Since exponents are multiplied, then (x 2) 5 = x 2 ∗ 5 = x 10 (\displaystyle (x^(2))^(5)=x^(2*5)=x^(10)). The point of this rule is that you multiply by powers (x 2) (\displaystyle (x^(2))) on itself five times. Like this:

      • (x 2) 5 (\displaystyle (x^(2))^(5))
      • (x 2) 5 = x 2 ∗ x 2 ∗ x 2 ∗ x 2 ∗ x 2 (\displaystyle (x^(2))^(5)=x^(2)*x^(2)*x^( 2)*x^(2)*x^(2))
      • Since the base is the same, the exponents simply add up: (x 2) 5 = x 2 ∗ x 2 ∗ x 2 ∗ x 2 ∗ x 2 = x 10 (\displaystyle (x^(2))^(5)=x^(2)*x^(2)* x^(2)*x^(2)*x^(2)=x^(10))

   3. A power with a negative exponent should be converted to a fraction (reverse power). It doesn't matter if you don't know what a reciprocal degree is. If you are given a degree with a negative exponent, e.g. 3 − 2 (\displaystyle 3^(-2)), write this degree in the denominator of the fraction (put 1 in the numerator), and make the exponent positive. In our example: 1 3 2 (\displaystyle (\frac (1)(3^(2)))). Here are other examples:

      When dividing degrees with the same base, their exponents are subtracted (the base does not change). The division operation is the opposite of the multiplication operation. For example, given the expression 4 4 4 2 (\displaystyle (\frac (4^(4))(4^(2)))). Subtract the exponent in the denominator from the exponent in the numerator (do not change the base). Thus, 4 4 4 2 = 4 4 − 2 = 4 2 (\displaystyle (\frac (4^(4))(4^(2)))=4^(4-2)=4^(2)) = 16 .

      • The power in the denominator can be written as follows: 1 4 2 (\displaystyle (\frac (1)(4^(2)))) = 4 − 2 (\displaystyle 4^(-2)). Remember that a fraction is a number (power, expression) with a negative exponent.

   4. Below are some expressions that will help you learn how to solve problems with exponents. The expressions given cover the material presented in this section. To see the answer, simply select the empty space after the equals sign.

      Solving problems with fractional exponents

      1. A power with a fractional exponent (for example, ) is converted to a root operation. In our example: x 1 2 (\displaystyle x^(\frac (1)(2))) = x (\displaystyle (\sqrt (x))). Here it does not matter what number is in the denominator of the fractional exponent. For example, x 1 4 (\displaystyle x^(\frac (1)(4)))- is the fourth root of “x”, that is x 4 (\displaystyle (\sqrt[(4)](x))) .

      2. If the exponent is improper fraction, then such a degree can be decomposed into two degrees to simplify the solution of the problem. There is nothing complicated about this - just remember the rule of multiplying powers. For example, a degree is given. Convert such a power into a root whose power is equal to the denominator of the fractional exponent, and then raise this root to a power equal to the numerator of the fractional exponent. To do this, remember that 5 3 (\displaystyle (\frac (5)(3))) = (1 3) ∗ 5 (\displaystyle ((\frac (1)(3)))*5). In our example:

         • x 5 3 (\displaystyle x^(\frac (5)(3)))
         • x 1 3 = x 3 (\displaystyle x^(\frac (1)(3))=(\sqrt[(3)](x)))
         • x 5 3 = x 5 ∗ x 1 3 (\displaystyle x^(\frac (5)(3))=x^(5)*x^(\frac (1)(3))) = (x 3) 5 (\displaystyle ((\sqrt[(3)](x)))^(5))
      3. Some calculators have a button to calculate exponents (you must first enter the base, then press the button, and then enter the exponent). It is denoted as ^ or x^y.
      4. Remember that any number to the first power is equal to itself, for example, 4 1 = 4. (\displaystyle 4^(1)=4.) Moreover, any number multiplied or divided by one is equal to itself, e.g. 5 ∗ 1 = 5 (\displaystyle 5*1=5) And 5 / 1 = 5 (\displaystyle 5/1=5).
      5. Know that the power 0 0 does not exist (such a power has no solution). If you try to solve such a degree on a calculator or on a computer, you will receive an error. But remember that any number in zero degree equals 1, for example, 4 0 = 1. (\displaystyle 4^(0)=1.)
      6. In higher mathematics, which operates with imaginary numbers: e a i x = c o s a x + i s i n a x (\displaystyle e^(a)ix=cosax+isinax), Where i = (− 1) (\displaystyle i=(\sqrt (())-1)); e is a constant approximately equal to 2.7; a is an arbitrary constant. The proof of this equality can be found in any textbook on higher mathematics.

      Warnings

      • As the exponent increases, its value increases greatly. So if the answer seems wrong to you, it may actually be correct. You can test this by graphing any exponential function, such as 2 x.

Convenient and simple online calculator fractions with detailed solutions Maybe:

• Add, subtract, multiply and divide fractions online,
• Receive ready-made solution fractions with a picture and it is convenient to transfer it.



The result of solving fractions will be here...

0 1 2 3 4 5 6 7 8 9
Fraction sign "/" + - * :
_erase Clear
Our online fraction calculator has quick input. To solve fractions, for example, simply write 1/2+2/7 into the calculator and press the " Solve fractions". The calculator will write to you detailed solution fractions and will issue an easy-to-copy image.

Signs used for writing in a calculator

You can type an example for a solution either from the keyboard or using buttons.

Features of the online fraction calculator

The fraction calculator can only perform operations on 2 simple fractions. They can be either correct(numerator less than the denominator), and incorrect (the numerator is greater than the denominator). The numbers in the numerator and denominators cannot be negative or greater than 999.
Our online calculator solves fractions and gives the answer to the right kind- reduces the fraction and selects the whole part, if necessary.

If you need to solve negative fractions, simply use the properties of minus. When multiplying and dividing negative fractions two negatives make an affirmative. That is, the product and division of negative fractions is equal to the product and division of the same positive ones. If one fraction is negative when multiplying or dividing, then simply remove the minus and then add it to the answer. When adding negative fractions, the result will be the same as if you were adding the same positive fractions. If you add one negative fraction, then this is the same as subtracting the same positive one.
When subtracting negative fractions, the result will be the same as if they were swapped and made positive. That is, minus by minus in this case gives a plus, but rearranging the terms does not change the sum. We use the same rules when subtracting fractions, one of which is negative.

To solve mixed fractions(fractions in which the whole part is highlighted) simply drive the whole part into the fraction. To do this, multiply the whole part by the denominator and add to the numerator.

If you need to solve 3 or more fractions online, you should solve them one by one. First, count the first 2 fractions, then solve the next fraction with the answer you get, and so on. Perform the operations one by one, 2 fractions at a time, and eventually you will get the correct answer.