In the course of studying algebra, we came across the concepts of polynomial (for example ($y-x$ ,$\ 2x^2-2x$ and so on) and algebraic fraction (for example $\frac(x+5)(x)$ , $\frac(2x ^2)(2x^2-2x)$,$\ \frac(x-y)(y-x)$ etc.) The similarity of these concepts is that both in polynomials and in algebraic fractions there are variables and numerical values, arithmetic actions: addition, subtraction, multiplication, exponentiation The difference between these concepts is that in polynomials division by a variable is not performed, and in algebraic fractions division by a variable can be performed.
Both polynomials and algebraic fractions are called rational algebraic expressions in mathematics. But polynomials are integer rational expressions, and algebraic fractional expressions are fractionally rational expressions.
It is possible to obtain a whole algebraic expression from a fractional-rational expression using the identical transformation, which in this case will be the main property of a fraction - reduction of fractions. Let's check it out in practice:
Example 1
Transform:$\ \frac(x^2-4x+4)(x-2)$
Solution: This fractional-rational equation can be transformed by using the basic property of the fraction-cancellation, i.e. dividing the numerator and denominator by the same number or expression other than $0$.
This fraction cannot be reduced immediately, it is necessary to convert the numerator.
We transform the expression in the numerator of the fraction, for this we use the formula for the square of the difference: $a^2-2ab+b^2=((a-b))^2$
The fraction has the form
\[\frac(x^2-4x+4)(x-2)=\frac(x^2-4x+4)(x-2)=\frac(((x-2))^2)( x-2)=\frac(\left(x-2\right)(x-2))(x-2)\]
Now we see that there is a common factor in the numerator and denominator - this is the expression $x-2$, on which we will reduce the fraction
\[\frac(x^2-4x+4)(x-2)=\frac(x^2-4x+4)(x-2)=\frac(((x-2))^2)( x-2)=\frac(\left(x-2\right)(x-2))(x-2)=x-2\]
After reduction, we have obtained that the original fractional-rational expression $\frac(x^2-4x+4)(x-2)$ has become a polynomial $x-2$, i.e. whole rational.
Now let's pay attention to the fact that the expressions $\frac(x^2-4x+4)(x-2)$ and $x-2\ $ can be considered identical not for all values of the variable, because in order for a fractional-rational expression to exist and for the reduction by the polynomial $x-2$ to be possible, the denominator of the fraction should not be equal to $0$ (as well as the factor by which we reduce. In this example, the denominator and factor are the same, but this is not always the case).
Variable values for which the algebraic fraction will exist are called valid variable values.
We put a condition on the denominator of the fraction: $x-2≠0$, then $x≠2$.
So the expressions $\frac(x^2-4x+4)(x-2)$ and $x-2$ are identical for all values of the variable except $2$.
Definition 1
identically equal Expressions are those that are equal for all possible values of the variable.
An identical transformation is any replacement of the original expression with an identically equal one. Such transformations include performing actions: addition, subtraction, multiplication, taking a common factor out of brackets, bringing algebraic fractions to common denominator, reduction of algebraic fractions, reduction of like terms, etc. It must be taken into account that a number of transformations, such as reduction, reduction of similar terms, can change the allowable values of the variable.
Techniques used to prove identities
Convert the left side of the identity to the right side or vice versa using identity transformations
Reduce both parts to the same expression using identical transformations
Transfer the expressions in one part of the expression to another and prove that the resulting difference is equal to $0$
Which of the above methods to use to prove a given identity depends on the original identity.
Example 2
Prove the identity $((a+b+c))^2- 2(ab+ac+bc)=a^2+b^2+c^2$
Solution: To prove this identity, we use the first of the above methods, namely, we will transform the left side of the identity until it is equal to the right side.
Consider the left side of the identity: $\ ((a+b+c))^2- 2(ab+ac+bc)$- it is the difference of two polynomials. In this case, the first polynomial is the square of the sum of three terms. To square the sum of several terms, we use the formula:
\[((a+b+c))^2=a^2+b^2+c^2+2ab+2ac+2bc\]
To do this, we need to multiply a number by a polynomial. Recall that for this we need to multiply the common factor outside the brackets by each term of the polynomial in brackets. Then we get:
$2(ab+ac+bc)=2ab+2ac+2bc$
Now back to the original polynomial, it will take the form:
$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)$
Note that there is a “-” sign in front of the bracket, which means that when the brackets are opened, all the signs that were in the brackets are reversed.
$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)= a ^2+b^2+c^2+2ab+2ac+2bc-2ab-2ac-2bc$
If we bring similar terms, then we get that the monomials $2ab$, $2ac$,$\ 2bc$ and $-2ab$,$-2ac$, $-2bc$ cancel each other out, i.e. their sum is equal to $0$.
$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)= a ^2+b^2+c^2+2ab+2ac+2bc-2ab-2ac-2bc=a^2+b^2+c^2$
So, by identical transformations, we obtained the identical expression on the left side of the original identity
$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2$
Note that the resulting expression shows that the original identity is true.
Note that in the original identity, all values of the variable are allowed, which means that we have proved the identity using identical transformations, and it is true for all allowed values of the variable.
Let two algebraic expressions be given:
Let's make a table of the values of each of these expressions for different numerical values of the letter x.
We see that for all those values that were given to the letter x, the values of both expressions turned out to be equal. The same will be true for any other value of x.
To verify this, we transform the first expression. Based on the distribution law, we write:
Having performed the indicated operations on the numbers, we get:
So, the first expression, after its simplification, turned out to be exactly the same as the second expression.
Now it is clear that for any value of x, the values of both expressions are equal.
Expressions whose values are equal for any values of the letters included in them are called identically equal or identical.
Hence, they are identical expressions.
Let's make one important remark. Let's take expressions:
Having compiled a table similar to the previous one, we will make sure that both expressions, for any value of x, except for have equal numerical values. Only when the second expression is equal to 6, and the first loses its meaning, since the denominator is zero. (Recall that you cannot divide by zero.) Can we say that these expressions are identical?
We agreed earlier that each expression will be considered only for admissible values of letters, that is, for those values for which the expression does not lose its meaning. This means that here, when comparing two expressions, we take into account only those letter values that are valid for both expressions. Therefore, we must exclude the value. And since for all other values of x both expressions have the same numerical value, we have the right to consider them identical.
Based on what has been said, we give the following definition of identical expressions:
1. Expressions are called identical if they have the same numerical values for all admissible values of the letters included in them.
If two identical expressions connect with an equal sign, we get an identity. Means:
2. An identity is an equality that is true for all admissible values of the letters included in it.
We have already encountered identities before. So, for example, all equalities are identities, with which we expressed the basic laws of addition and multiplication.
For example, equalities expressing the commutative law of addition
and the associative law of multiplication
are valid for any values of letters. Hence, these equalities are identities.
All true arithmetic equalities are also considered identities, for example:
In algebra, one often has to replace an expression with another that is identical to it. Let, for example, it is required to find the value of the expression
We will greatly facilitate the calculations if we replace the given expression with an expression that is identical to it. Based on the distribution law, we can write:
But the numbers in brackets add up to 100. So, we have an identity:
Substituting 6.53 instead of a on the right side of it, we immediately (in the mind) find the numerical value (653) of this expression.
Replacing one expression with another, identical to it, is called the identical transformation of this expression.
Recall that any algebraic expression for any admissible values of letters is some
number. It follows from this that all the laws and properties of arithmetic operations that were given in the previous chapter are applicable to algebraic expressions. So, the application of the laws and properties of arithmetic operations transforms a given algebraic expression into an expression that is identical to it.
Identity transformations are the work we do with numerical and literal expressions, as well as with expressions that contain variables. We carry out all these transformations in order to bring the original expression to a form that will be convenient for solving the problem. We will consider the main types of identical transformations in this topic.
Identity transformation of an expression. What it is?
For the first time we meet with the concept of identical transformed we in algebra lessons in grade 7. Then we first get acquainted with the concept of identically equal expressions. Let's deal with the concepts and definitions to facilitate the assimilation of the topic.
Definition 1
Identity transformation of an expression are actions performed to replace the original expression with an expression that will be identically equal to the original one.
Often this definition is used in an abbreviated form, in which the word "identical" is omitted. It is assumed that in any case we carry out the transformation of the expression in such a way as to obtain an expression identical to the original one, and this does not need to be emphasized separately.
Illustrate this definition examples.
Example 1
If we replace the expression x + 3 - 2 to the identically equal expression x+1, then we carry out the identical transformation of the expression x + 3 - 2.
Example 2
Replacing expression 2 a 6 with expression a 3 is the identity transformation, while the replacement of the expression x to the expression x2 is not an identical transformation, since the expressions x and x2 are not identically equal.
We draw your attention to the form of writing expressions when carrying out identical transformations. We usually write the original expression and the resulting expression as an equality. So, writing x + 1 + 2 = x + 3 means that the expression x + 1 + 2 has been reduced to the form x + 3 .
Sequential execution of actions leads us to a chain of equalities, which is several consecutive identical transformations. So, we understand the notation x + 1 + 2 = x + 3 = 3 + x as a sequential implementation of two transformations: first, the expression x + 1 + 2 was reduced to the form x + 3, and it was reduced to the form 3 + x.
Identity transformations and ODZ
A number of expressions that we begin to study in grade 8 do not make sense for any values of variables. Carrying out identical transformations in these cases requires us to pay attention to the region of admissible values of variables (ODV). Performing identical transformations may leave the ODZ unchanged or narrow it down.
Example 3
When performing a transition from the expression a + (−b) to the expression a-b range of allowed values of variables a and b stays the same.
Example 4
Transition from expression x to expression x 2 x leads to a narrowing of the range of admissible values of the variable x from the set of all real numbers to the set of all real numbers from which zero has been excluded.
Example 5
Identity transformation of an expression x 2 x expression x leads to the expansion of the range of valid values of the variable x from the set of all real numbers except for zero to the set of all real numbers.
Narrowing or expanding the range of allowable values of variables when carrying out identical transformations is important in solving problems, since it can affect the accuracy of calculations and lead to errors.
Basic identity transformations
Let's now see what identical transformations are and how they are performed. Let us single out those types of identical transformations that we have to deal with most often into the main group.
In addition to the basic identity transformations, there are a number of transformations that relate to expressions of a particular type. For fractions, these are methods of reduction and reduction to a new denominator. For expressions with roots and powers, all actions that are performed based on the properties of roots and powers. For logarithmic expressions, actions that are performed based on the properties of logarithms. For trigonometric expressions all actions using trigonometric formulas. All these particular transformations are discussed in detail in selected topics which can be found on our website. For this reason, we will not dwell on them in this article.
Let us proceed to the consideration of the main identical transformations.
Rearrangement of terms, factors
Let's start by rearranging the terms. We deal with this identical transformation most often. And the following statement can be considered the main rule here: in any sum, the rearrangement of the terms in places does not affect the result.
This rule is based on the commutative and associative properties of addition. These properties allow us to rearrange the terms in places and at the same time obtain expressions that are identically equal to the original ones. That is why the rearrangement of terms in places in the sum is an identical transformation.
Example 6
We have the sum of three terms 3 + 5 + 7 . If we swap the terms 3 and 5, then the expression will take the form 5 + 3 + 7. There are several options for rearranging the terms in this case. All of them lead to obtaining expressions that are identically equal to the original one.
Not only numbers, but also expressions can act as terms in the sum. They, just like numbers, can be rearranged without affecting the final result of calculations.
Example 7
In the sum of three terms 1 a + b, a 2 + 2 a + 5 + a 7 a 3 and - 12 a of the form 1 a + b + a 2 + 2 a + 5 + a 7 a 3 + ( - 12) a terms can be rearranged, for example, like this (- 12) a + 1 a + b + a 2 + 2 a + 5 + a 7 a 3 . In turn, you can rearrange the terms in the denominator of the fraction 1 a + b, while the fraction will take the form 1 b + a. And the expression under the root sign a 2 + 2 a + 5 is also a sum in which the terms can be interchanged.
In the same way as the terms, in the original expressions one can interchange the factors and obtain identically correct equations. This action is governed by the following rule:
Definition 2
In the product, rearranging the factors in places does not affect the result of the calculation.
This rule is based on the commutative and associative properties of multiplication, which confirm the correctness of the identical transformation.
Example 8
Work 3 5 7 permutation of factors can be represented in one of the following forms: 5 3 7 , 5 7 3 , 7 3 5 , 7 5 3 or 3 7 5.
Example 9
Permuting the factors in the product x + 1 x 2 - x + 1 x will give x 2 - x + 1 x x + 1
Bracket expansion
Parentheses can contain entries of numeric expressions and expressions with variables. These expressions can be transformed into identically equal expressions, in which there will be no parentheses at all or there will be fewer of them than in the original expressions. This way of converting expressions is called parenthesis expansion.
Example 10
Let's carry out actions with brackets in an expression of the form 3 + x − 1 x in order to get the identically true expression 3 + x − 1 x.
The expression 3 · x - 1 + - 1 + x 1 - x can be converted to the identically equal expression without brackets 3 · x - 3 - 1 + x 1 - x .
We discussed in detail the rules for converting expressions with brackets in the topic "Bracket expansion", which is posted on our resource.
Grouping terms, factors
In cases where we are dealing with three or more terms, we can resort to such a type of identical transformations as a grouping of terms. By this method of transformation is meant the union of several terms into a group by rearranging them and placing them in brackets.
When grouping, the terms are interchanged in such a way that the grouped terms are in the expression record next to each other. After that, they can be enclosed in brackets.
Example 11
Take the expression 5 + 7 + 1 . If we group the first term with the third, we get (5 + 1) + 7 .
The grouping of factors is carried out similarly to the grouping of terms.
Example 12
In the work 2 3 4 5 it is possible to group the first factor with the third, and the second factor with the fourth, in this case we arrive at the expression (2 4) (3 5). And if we grouped the first, second and fourth factors, we would get the expression (2 3 5) 4.
The terms and factors that are grouped can be represented as prime numbers, as well as expressions. The grouping rules were discussed in detail in the topic "Grouping terms and factors".
Replacing differences by sums, partial products and vice versa
The replacement of differences by sums became possible thanks to our acquaintance with opposite numbers. Now subtraction from a number a numbers b can be seen as an addition to the number a numbers −b. Equality a − b = a + (− b) can be considered fair and, on its basis, carry out the replacement of differences by sums.
Example 13
Take the expression 4 + 3 − 2 , in which the difference of numbers 3 − 2 we can write as the sum 3 + (− 2) . Get 4 + 3 + (− 2) .
Example 14
All differences in expression 5 + 2 x - x 2 - 3 x 3 - 0, 2 can be replaced by sums like 5 + 2 x + (− x 2) + (− 3 x 3) + (− 0 , 2).
We can proceed to sums from any differences. Similarly, we can make a reverse substitution.
The replacement of division by multiplication by the reciprocal of the divisor is made possible by the concept of reciprocal numbers. This transformation can be written as a: b = a (b − 1).
This rule was the basis of the rule for dividing ordinary fractions.
Example 15
Private 1 2: 3 5 can be replaced by a product of the form 1 2 5 3.
Similarly, by analogy, division can be replaced by multiplication.
Example 16
In the case of the expression 1+5:x:(x+3) replace division with x can be multiplied by 1 x. Division by x + 3 we can replace by multiplying with 1 x + 3. The transformation allows us to obtain an expression that is identical to the original one: 1 + 5 1 x 1 x + 3 .
Replacing multiplication by division is carried out according to the scheme a b = a: (b − 1).
Example 17
In the expression 5 x x 2 + 1 - 3, multiplication can be replaced by division as 5: x 2 + 1 x - 3.
Performing actions with numbers
Performing operations with numbers is subject to the rule of order of operations. First, operations are performed with powers of numbers and roots of numbers. After that, we replace logarithms, trigonometric and other functions with their values. Then the actions in parentheses are performed. And then you can already carry out all the other actions from left to right. It is important to remember that multiplication and division are carried out before addition and subtraction.
Operations with numbers allow you to transform the original expression into an identical one equal to it.
Example 18
Let's transform the expression 3 · 2 3 - 1 · a + 4 · x 2 + 5 · x by performing all possible operations with numbers.
Solution
First, let's look at the degree 2 3 and root 4 and calculate their values: 2 3 = 8 and 4 = 2 2 = 2 .
Substitute the obtained values into the original expression and get: 3 (8 - 1) a + 2 (x 2 + 5 x) .
Now let's do the parentheses: 8 − 1 = 7 . And let's move on to the expression 3 7 a + 2 (x 2 + 5 x) .
We just have to do the multiplication 3 and 7 . We get: 21 a + 2 (x 2 + 5 x) .
Answer: 3 2 3 - 1 a + 4 x 2 + 5 x = 21 a + 2 (x 2 + 5 x)
Operations with numbers may be preceded by other kinds of identity transformations, such as number grouping or parenthesis expansion.
Example 19
Take the expression 3 + 2 (6: 3) x (y 3 4) − 2 + 11.
Solution
First of all, we will change the quotient in parentheses 6: 3 on its meaning 2 . We get: 3 + 2 2 x (y 3 4) − 2 + 11 .
Let's expand the brackets: 3 + 2 2 x (y 3 4) − 2 + 11 = 3 + 2 2 x y 3 4 − 2 + 11.
Let's group numerical factors in the product, as well as terms that are numbers: (3 − 2 + 11) + (2 2 4) x y 3.
Let's do the parentheses: (3 − 2 + 11) + (2 2 4) x y 3 = 12 + 16 x y 3
Answer:3 + 2 (6: 3) x (y 3 4) − 2 + 11 = 12 + 16 x y 3
If we work with numerical expressions, then the purpose of our work will be to find the value of the expression. If we transform expressions with variables, then the goal of our actions will be to simplify the expression.
Bracketing the Common Factor
In cases where the terms in the expression have the same factor, then we can take this common factor out of brackets. To do this, we first need to represent the original expression as the product of a common factor and an expression in brackets, which consists of the original terms without a common factor.
Example 20
Numerically 2 7 + 2 3 we can take out the common factor 2 outside the brackets and get an identically correct expression of the form 2 (7 + 3).
You can refresh the memory of the rules for putting the common factor out of brackets in the corresponding section of our resource. The material discusses in detail the rules for taking the common factor out of brackets and provides numerous examples.
Reduction of similar terms
Now let's move on to sums that contain like terms. Two options are possible here: sums containing the same terms, and sums whose terms differ by a numerical coefficient. Operations with sums containing like terms is called reduction of like terms. It is carried out as follows: we put the common letter part out of brackets and calculate the sum of numerical coefficients in brackets.
Example 21
Consider the expression 1 + 4 x − 2 x. We can take the literal part of x out of brackets and get the expression 1 + x (4 − 2). Let's calculate the value of the expression in brackets and get the sum of the form 1 + x · 2 .
Replacing numbers and expressions with identically equal expressions
The numbers and expressions that make up the original expression can be replaced by expressions that are identically equal to them. Such a transformation of the original expression leads to an expression that is identically equal to it.
Example 22 Example 23
Consider the expression 1 + a5, in which we can replace the degree a 5 with a product identically equal to it, for example, of the form a 4. This will give us the expression 1 + a 4.
The transformation performed is artificial. It only makes sense in preparation for other transformations.
Example 24
Consider the transformation of the sum 4 x 3 + 2 x 2. Here the term 4x3 we can represent as a product 2 x 2 x 2 x. As a result, the original expression takes the form 2 x 2 2 x + 2 x 2. Now we can isolate the common factor 2x2 and take it out of the brackets: 2 x 2 (2 x + 1).
Adding and subtracting the same number
Adding and subtracting the same number or expression at the same time is an artificial expression transformation technique.
Example 25
Consider the expression x 2 + 2 x. We can add or subtract one from it, which will allow us to subsequently carry out another identical transformation - to select the square of the binomial: x 2 + 2 x = x 2 + 2 x + 1 - 1 = (x + 1) 2 - 1.
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The arithmetic operation that is performed last when calculating the value of the expression is the "main".
That is, if you substitute some (any) numbers instead of letters, and try to calculate the value of the expression, then if the last action is multiplication, then we have a product (the expression is decomposed into factors).
If the last action is addition or subtraction, this means that the expression is not factored (and therefore cannot be reduced).
To fix it yourself, a few examples:
Examples:
Solutions:
1. I hope you did not immediately rush to cut and? It was still not enough to “reduce” units like this:
The first step should be to factorize:
4. Addition and subtraction of fractions. Bringing fractions to a common denominator.
Adding and subtracting ordinary fractions is a well-known operation: we look for a common denominator, multiply each fraction by the missing factor and add / subtract the numerators.
Let's remember:
Answers:
1. The denominators and are coprime, that is, they do not have common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:
2. Here the common denominator is:
3. First thing here mixed fractions turn them into wrong ones, and then - according to the usual scheme:
It is quite another matter if the fractions contain letters, for example:
Let's start simple:
a) Denominators do not contain letters
Here everything is the same as with ordinary numerical fractions: we find a common denominator, multiply each fraction by the missing factor and add / subtract the numerators:
now in the numerator you can bring similar ones, if any, and factor them:
Try it yourself:
Answers:
b) Denominators contain letters
Let's remember the principle of finding a common denominator without letters:
First of all, we define common factors;
Then we write out all the common factors once;
and multiply them by all other factors, not common ones.
To determine the common factors of the denominators, we first decompose them into simple factors:
We emphasize the common factors:
Now we write out the common factors once and add to them all non-common (not underlined) factors:
This is the common denominator.
Let's get back to the letters. The denominators are given in exactly the same way:
We decompose the denominators into factors;
determine common (identical) multipliers;
write out all the common factors once;
We multiply them by all other factors, not common ones.
So, in order:
1) decompose the denominators into factors:
2) determine the common (identical) factors:
3) write out all the common factors once and multiply them by all the other (not underlined) factors:
So the common denominator is here. The first fraction must be multiplied by, the second - by:
By the way, there is one trick:
For example: .
We see the same factors in the denominators, only all with different indicators. The common denominator will be:
to the extent
to the extent
to the extent
in degree.
Let's complicate the task:
How to make fractions have the same denominator?
Let's remember the basic property of a fraction:
Nowhere is it said that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because it's not true!
See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example, . What has been learned?
So, another unshakable rule:
When you bring fractions to a common denominator, use only the multiplication operation!
But what do you need to multiply to get?
Here on and multiply. And multiply by:
Expressions that cannot be factorized will be called "elementary factors".
For example, is an elementary factor. - too. But - no: it is decomposed into factors.
What about expression? Is it elementary?
No, because it can be factorized:
(you already read about factorization in the topic "").
So, the elementary factors into which you decompose the expression with letters is an analogue prime factors into which you decompose the numbers. And we will do the same with them.
We see that both denominators have a factor. It will go to the common denominator in the power (remember why?).
The multiplier is elementary, and they do not have it in common, which means that the first fraction will simply have to be multiplied by it:
Another example:
Solution:
Before multiplying these denominators in a panic, you need to think about how to factor them? Both of them represent:
Excellent! Then:
Another example:
Solution:
As usual, we factorize the denominators. In the first denominator, we simply put it out of brackets; in the second - the difference of squares:
It would seem that there are no common factors. But if you look closely, they are already so similar ... And the truth is:
So let's write:
That is, it turned out like this: inside the bracket, we swapped the terms, and at the same time, the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.
Now we bring to a common denominator:
Got it? Now let's check.
Tasks for independent solution:
Answers:
Here we must remember one more thing - the difference of cubes:
Please note that the denominator of the second fraction does not contain the formula "square of the sum"! The square of the sum would look like this:
A is the so-called incomplete square of the sum: the second term in it is the product of the first and last, and not their doubled product. The incomplete square of the sum is one of the factors in the expansion of the difference of cubes:
What if there are already three fractions?
Yes, the same! First of all, let's make it so that maximum amount factors in the denominators were the same:
Pay attention: if you change the signs inside one bracket, the sign in front of the fraction changes to the opposite. When we change the signs in the second bracket, the sign in front of the fraction is reversed again. As a result, he (the sign in front of the fraction) has not changed.
We write out the first denominator in full in the common denominator, and then we add to it all the factors that have not yet been written, from the second, and then from the third (and so on, if there are more fractions). That is, it goes like this:
Hmm ... With fractions, it’s clear what to do. But what about the two?
It's simple: you know how to add fractions, right? So, you need to make sure that the deuce becomes a fraction! Remember: a fraction is a division operation (the numerator is divided by the denominator, in case you suddenly forgot). And there is nothing easier than dividing a number by. In this case, the number itself will not change, but will turn into a fraction:
Exactly what is needed!
5. Multiplication and division of fractions.
Well, the hardest part is now over. And ahead of us is the simplest, but at the same time the most important:
Procedure
What is the procedure for counting numeric expression? Remember, considering the value of such an expression:
Did you count?
It should work.
So, I remind you.
The first step is to calculate the degree.
The second is multiplication and division. If there are several multiplications and divisions at the same time, you can do them in any order.
And finally, we perform addition and subtraction. Again, in any order.
But: the parenthesized expression is evaluated out of order!
If several brackets are multiplied or divided by each other, we first evaluate the expression in each of the brackets, and then multiply or divide them.
What if there are other parentheses inside the brackets? Well, let's think: some expression is written inside the brackets. What is the first thing to do when evaluating an expression? That's right, calculate brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.
So, the order of actions for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):
Okay, it's all simple.
But that's not the same as an expression with letters, is it?
No, it's the same! Only instead of arithmetic operations it is necessary to do algebraic operations, that is, the operations described in the previous section: bringing similar, adding fractions, reducing fractions, and so on. The only difference will be the action of factoring polynomials (we often use it when working with fractions). Most often, for factorization, you need to use i or simply take the common factor out of brackets.
Usually our goal is to represent an expression as a product or quotient.
For example:
Let's simplify the expression.
1) First we simplify the expression in brackets. There we have the difference of fractions, and our goal is to represent it as a product or quotient. So, we bring the fractions to a common denominator and add:
It is impossible to simplify this expression further, all factors here are elementary (do you still remember what this means?).
2) We get:
Multiplication of fractions: what could be easier.
3) Now you can shorten:
OK it's all over Now. Nothing complicated, right?
Another example:
Simplify the expression.
First, try to solve it yourself, and only then look at the solution.
Solution:
First of all, let's define the procedure.
First, let's add the fractions in brackets, instead of two fractions, one will turn out.
Then we will do the division of fractions. Well, we add the result with the last fraction.
I will schematically number the steps:
Now I will show the whole process, tinting the current action with red:
1. If there are similar ones, they must be brought immediately. At whatever moment we have similar ones, it is advisable to bring them right away.
2. The same goes for reducing fractions: as soon as an opportunity arises to reduce, it must be used. The exception is fractions that you add or subtract: if they now have the same denominators, then the reduction should be left for later.
Here are some tasks for you to solve on your own:
And promised at the very beginning:
Answers:
Solutions (brief):
If you coped with at least the first three examples, then you, consider, have mastered the topic.
Now on to learning!
EXPRESSION CONVERSION. SUMMARY AND BASIC FORMULA
Basic simplification operations:
- Bringing similar: to add (reduce) like terms, you need to add their coefficients and assign the letter part.
- Factorization: taking the common factor out of brackets, applying, etc.
- Fraction reduction: the numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, from which the value of the fraction does not change.
1) numerator and denominator factorize
2) if there are common factors in the numerator and denominator, they can be crossed out.IMPORTANT: only multipliers can be reduced!
- Addition and subtraction of fractions:
; - Multiplication and division of fractions:
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