This article continues the theme of transformation algebraic fractions: consider such an action as reducing algebraic fractions. Let's define the term itself, formulate a reduction rule and analyze practical examples.
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The meaning of reducing an algebraic fraction
In materials about common fractions, we looked at its reduction. We defined reducing a fraction as dividing its numerator and denominator by a common factor.
Reducing an algebraic fraction is a similar operation.
Definition 1
Reducing an algebraic fraction is the division of its numerator and denominator by a common factor. In this case, in contrast to the reduction of an ordinary fraction (the common denominator can only be a number), the common factor of the numerator and denominator of an algebraic fraction can be a polynomial, in particular, a monomial or a number.
For example, the algebraic fraction 3 x 2 + 6 x y 6 x 3 y + 12 x 2 y 2 can be reduced by the number 3, resulting in: x 2 + 2 x y 6 x 3 · y + 12 · x 2 · y 2 . We can reduce the same fraction by the variable x, and this will give us the expression 3 x + 6 y 6 x 2 y + 12 x y 2. It is also possible to reduce a given fraction by a monomial 3 x or any of the polynomials x + 2 y, 3 x + 6 y , x 2 + 2 x y or 3 x 2 + 6 x y.
The ultimate goal of reducing an algebraic fraction is a fraction greater than simple type, at best, is an irreducible fraction.
Are all algebraic fractions subject to reduction?
Again, from materials on ordinary fractions, we know that there are reducible and irreducible fractions. Irreducible fractions are fractions that do not have common numerator and denominator factors other than 1.
It’s the same with algebraic fractions: they may have common factors in the numerator and denominator, or they may not. The presence of common factors allows you to simplify the original fraction through reduction. When there are no common factors, it is impossible to optimize a given fraction using the reduction method.
In general cases, given the type of fraction it is quite difficult to understand whether it can be reduced. Of course, in some cases the presence of a common factor between the numerator and denominator is obvious. For example, in the algebraic fraction 3 x 2 3 y it is clear that the common factor is the number 3.
In the fraction - x · y 5 · x · y · z 3 we also immediately understand that it can be reduced by x, or y, or x · y. And yet, much more often there are examples of algebraic fractions, when the common factor of the numerator and denominator is not so easy to see, and even more often, it is simply absent.
For example, we can reduce the fraction x 3 - 1 x 2 - 1 by x - 1, while the specified common factor is not present in the entry. But the fraction x 3 - x 2 + x - 1 x 3 + x 2 + 4 · x + 4 cannot be reduced, since the numerator and denominator do not have a common factor.
Thus, the question of determining the reducibility of an algebraic fraction is not so simple, and it is often easier to work with a fraction of a given form than to try to find out whether it is reducible. In this case, such transformations take place that in particular cases make it possible to determine the common factor of the numerator and denominator or to draw a conclusion about the irreducibility of a fraction. We will examine this issue in detail in the next paragraph of the article.
Rule for reducing algebraic fractions
Rule for reducing algebraic fractions consists of two sequential actions:
- finding common factors of the numerator and denominator;
- if any are found, the action of reducing the fraction is carried out directly.
The most convenient method of finding common denominators is to factor the polynomials present in the numerator and denominator of a given algebraic fraction. This allows you to immediately clearly see the presence or absence of common factors.
The very action of reducing an algebraic fraction is based on the main property of an algebraic fraction, expressed by the equality undefined, where a, b, c are some polynomials, and b and c are non-zero. The first step is to reduce the fraction to the form a · c b · c, in which we immediately notice the common factor c. The second step is to perform a reduction, i.e. transition to a fraction of the form a b .
Typical examples
Despite some obviousness, let us clarify the special case when the numerator and denominator of an algebraic fraction are equal. Similar fractions are identically equal to 1 on the entire ODZ of the variables of this fraction:
5 5 = 1 ; - 2 3 - 2 3 = 1 ; x x = 1 ; - 3, 2 x 3 - 3, 2 x 3 = 1; 1 2 · x - x 2 · y 1 2 · x - x 2 · y ;
Since ordinary fractions are a special case of algebraic fractions, let us recall how they are reduced. The natural numbers written in the numerator and denominator are factored into prime factors, then the common factors are canceled (if any).
For example, 24 1260 = 2 2 2 3 2 2 3 3 5 7 = 2 3 5 7 = 2 105
The product of simple identical factors can be written as powers, and in the process of reducing a fraction, use the property of dividing powers with identical bases. Then the above solution would be:
24 1260 = 2 3 3 2 2 3 2 5 7 = 2 3 - 2 3 2 - 1 5 7 = 2 105
(numerator and denominator divided by a common factor 2 2 3). Or for clarity, based on the properties of multiplication and division, we give the solution the following form:
24 1260 = 2 3 3 2 2 3 2 5 7 = 2 3 2 2 3 3 2 1 5 7 = 2 1 1 3 1 35 = 2 105
By analogy, the reduction of algebraic fractions is carried out, in which the numerator and denominator have monomials with integer coefficients.
Example 1
The algebraic fraction is given - 27 · a 5 · b 2 · c · z 6 · a 2 · b 2 · c 7 · z. It needs to be reduced.
Solution
It is possible to write the numerator and denominator of a given fraction as a product of simple factors and variables, and then carry out the reduction:
27 · a 5 · b 2 · c · z 6 · a 2 · b 2 · c 7 · z = - 3 · 3 · 3 · a · a · a · a · a · b · b · c · z 2 · 3 · a · a · b · b · c · c · c · c · c · c · c · z = = - 3 · 3 · a · a · a 2 · c · c · c · c · c · c = - 9 a 3 2 c 6
However, a more rational way would be to write the solution as an expression with powers:
27 · a 5 · b 2 · c · z 6 · a 2 · b 2 · c 7 · z = - 3 3 · a 5 · b 2 · c · z 2 · 3 · a 2 · b 2 · c 7 · z = - 3 3 2 · 3 · a 5 a 2 · b 2 b 2 · c c 7 · z z = = - 3 3 - 1 2 · a 5 - 2 1 · 1 · 1 c 7 - 1 · 1 = · - 3 2 · a 3 2 · c 6 = · - 9 · a 3 2 · c 6 .
Answer:- 27 a 5 b 2 c z 6 a 2 b 2 c 7 z = - 9 a 3 2 c 6
When the numerator and denominator of an algebraic fraction contain fractional numerical coefficients, there are two possible ways of further action: either divide these fractional coefficients separately, or first get rid of the fractional coefficients by multiplying the numerator and denominator by a certain natural number. The last transformation is carried out due to the basic property of an algebraic fraction (you can read about it in the article “Reducing an algebraic fraction to a new denominator”).
Example 2
The given fraction is 2 5 x 0, 3 x 3. It needs to be reduced.
Solution
It is possible to reduce the fraction this way:
2 5 x 0, 3 x 3 = 2 5 3 10 x x 3 = 4 3 1 x 2 = 4 3 x 2
Let's try to solve the problem differently, having first gotten rid of fractional coefficients - multiply the numerator and denominator by the least common multiple of the denominators of these coefficients, i.e. on LCM (5, 10) = 10. Then we get:
2 5 x 0, 3 x 3 = 10 2 5 x 10 0, 3 x 3 = 4 x 3 x 3 = 4 3 x 2.
Answer: 2 5 x 0, 3 x 3 = 4 3 x 2
When we reduce algebraic fractions general view, in which the numerators and denominators can be either monomials or polynomials, there may be a problem when the common factor is not always immediately visible. Or moreover, it simply does not exist. Then, to determine the common factor or record the fact of its absence, the numerator and denominator of the algebraic fraction are factored.
Example 3
The rational fraction 2 · a 2 · b 2 + 28 · a · b 2 + 98 · b 2 a 2 · b 3 - 49 · b 3 is given. It needs to be reduced.
Solution
Let us factor the polynomials in the numerator and denominator. Let's take it out of brackets:
2 a 2 b 2 + 28 a b 2 + 98 b 2 a 2 b 3 - 49 b 3 = 2 b 2 (a 2 + 14 a + 49) b 3 (a 2 - 49)
We see that the expression in parentheses can be converted using abbreviated multiplication formulas:
2 b 2 (a 2 + 14 a + 49) b 3 (a 2 - 49) = 2 b 2 (a + 7) 2 b 3 (a - 7) (a + 7)
It is clearly seen that it is possible to reduce a fraction by a common factor b 2 (a + 7). Let's make a reduction:
2 b 2 (a + 7) 2 b 3 (a - 7) (a + 7) = 2 (a + 7) b (a - 7) = 2 a + 14 a b - 7 b
Let us write a short solution without explanation as a chain of equalities:
2 a 2 b 2 + 28 a b 2 + 98 b 2 a 2 b 3 - 49 b 3 = 2 b 2 (a 2 + 14 a + 49) b 3 (a 2 - 49) = = 2 b 2 (a + 7) 2 b 3 (a - 7) (a + 7) = 2 (a + 7) b (a - 7) = 2 a + 14 a b - 7 b
Answer: 2 a 2 b 2 + 28 a b 2 + 98 b 2 a 2 b 3 - 49 b 3 = 2 a + 14 a b - 7 b.
It happens that common factors are hidden by numerical coefficients. Then, when reducing fractions, it is optimal to put the numerical factors at higher powers of the numerator and denominator out of brackets.
Example 4
Given the algebraic fraction 1 5 · x - 2 7 · x 3 · y 5 · x 2 · y - 3 1 2 . It is necessary to reduce it if possible.
Solution
At first glance, the numerator and denominator do not exist common denominator. However, let's try to convert the given fraction. Let's take out the factor x in the numerator:
1 5 x - 2 7 x 3 y 5 x 2 y - 3 1 2 = x 1 5 - 2 7 x 2 y 5 x 2 y - 3 1 2
Now you can see some similarity between the expression in brackets and the expression in the denominator due to x 2 y . Let us take out the numerical coefficients of the higher powers of these polynomials:
x 1 5 - 2 7 x 2 y 5 x 2 y - 3 1 2 = x - 2 7 - 7 2 1 5 + x 2 y 5 x 2 y - 1 5 3 1 2 = = - 2 7 x - 7 10 + x 2 y 5 x 2 y - 7 10
Now the common factor becomes visible, we carry out the reduction:
2 7 x - 7 10 + x 2 y 5 x 2 y - 7 10 = - 2 7 x 5 = - 2 35 x
Answer: 1 5 x - 2 7 x 3 y 5 x 2 y - 3 1 2 = - 2 35 x .
Let us emphasize that the skill of reducing rational fractions depends on the ability to factor polynomials.
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Fractions and their reduction is another topic that begins in 5th grade. Here the basis of this action is formed, and then these skills are drawn by a thread into higher mathematics. If the student does not understand, then he may have problems in algebra. Therefore, it is better to understand a few rules once and for all. And also remember one prohibition and never break it.
Fraction and its reduction
Every student knows what it is. Any two digits located between a horizontal line are immediately perceived as a fraction. However, not everyone understands that any number can become it. If it is an integer, then it can always be divided by one, and then you get an improper fraction. But more on that later.
The beginning is always simple. First you need to figure out how to reduce a proper fraction. That is, one in which the numerator is less than the denominator. To do this, you will need to remember the basic property of a fraction. It states that when multiplying (as well as dividing) its numerator and denominator at the same time by the same number, an equivalent fraction is obtained.
Division actions that are performed on this property and result in a reduction. That is, to simplify it as much as possible. A fraction can be reduced as long as there are common factors above and below the line. When they are no longer there, reduction is impossible. And they say that this fraction is irreducible.
Two ways
1.Step by step reduction. It uses an estimation method where both numbers are divided by the minimum common factor that the student notices. If after the first contraction it is clear that this is not the end, then the division continues. Until the fraction becomes irreducible.
2. Finding the greatest common divisor of the numerator and denominator. This is the most rational way to reduce fractions. It involves factoring the numerator and denominator into prime factors. Among them, you then need to choose all the same ones. Their product will give the greatest common factor by which the fraction is reduced.
Both of these methods are equivalent. The student is encouraged to master them and use the one he likes best.
What if there are letters and addition and subtraction operations?
The first part of the question is more or less clear. Letters can be abbreviated just like numbers. The main thing is that they act as multipliers. But many people have problems with the second one.
Important to remember! You can only reduce numbers that are factors. If they are summands, it is impossible.
In order to understand how to reduce fractions that have the form of an algebraic expression, you need to understand the rule. First, express the numerator and denominator as a product. Then you can reduce if common factors appear. To represent it in the form of multipliers, the following techniques are useful:
- grouping;
- bracketing;
- application of abbreviated multiplication identities.
Moreover, the latter method makes it possible to immediately obtain the terms in the form of multipliers. Therefore, it should always be used if a known pattern is visible.
But this is not scary yet, then tasks with degrees and roots appear. That's when you need to gain courage and learn a couple of new rules.
Expression with degree
Fraction. The numerator and denominator are the product. There are letters and numbers. And they are also raised to a power, which also consists of terms or factors. There is something to be afraid of.
In order to understand how to reduce fractions with powers, you will need to learn two things:
- if the exponent contains a sum, then it can be decomposed into factors, the powers of which will be the original terms;
- if the difference, then the dividend and the divisor, the first will have the minuend to the power, the second will have the subtrahend.
After completing these steps, the total multipliers become visible. In such examples there is no need to calculate all powers. It is enough to simply reduce degrees with the same exponents and bases.
In order to finally master how to reduce fractions with powers, you need a lot of practice. After several similar examples, actions will be performed automatically.
What if the expression contains a root?
It can also be shortened. Only again, following the rules. Moreover, all those described above are true. In general, if the question is how to reduce a fraction with roots, then you need to divide.
It can also be divided into irrational expressions. That is, if the numerator and denominator have identical factors, enclosed under the sign of the root, then they can be safely reduced. This will simplify the expression and complete the task.
If, after the reduction, irrationality remains under the fraction line, then you need to get rid of it. In other words, multiply the numerator and denominator by it. If common factors appear after this operation, they will need to be reduced again.
That's probably all about how to reduce fractions. There are few rules, but only one ban. Never shorten terms!
Children at school learn the rules of reducing fractions in 6th grade. In this article, we will first tell you what this action means, then we will explain how to convert a reducible fraction into an irreducible fraction. The next point will be the rules for reducing fractions, and then we will gradually get to the examples.
What does it mean to "reduce a fraction"?
So we all know that ordinary fractions are divided into two groups: reducible and irreducible. Already by the names you can understand that those that are contractible are contracted, and those that are irreducible are not contracted.
- To reduce a fraction means to divide its denominator and numerator by their (other than one) positive divisor. The result, of course, is new fraction with a smaller denominator and numerator. The resulting fraction will be equal to the original fraction.
It is worth noting that in mathematics books with the task “reduce a fraction,” this means that you need to reduce the original fraction to this irreducible form. If we talk in simple words, then dividing the denominator and numerator by their greatest common divisor is a reduction.
How to reduce a fraction. Rules for reducing fractions (grade 6)
So there are only two rules here.
- The first rule of reducing fractions is to first find the greatest common factor of the denominator and numerator of your fraction.
- The second rule: divide the denominator and numerator by the greatest common divisor, ultimately obtaining an irreducible fraction.
How to reduce an improper fraction?
The rules for reducing fractions are identical to the rules for reducing improper fractions.
In order to reduce an improper fraction, you will first need to factor the denominator and numerator into prime factors, and only then reduce the common factors.
Reducing mixed fractions
The rules for reducing fractions also apply to reducing mixed fractions. There is only a small difference: we can not touch the whole part, but reduce the fraction or convert the mixed fraction into an improper fraction, then reduce it and again convert it into a proper fraction.
There are two ways to reduce mixed fractions.
First: write the fractional part into prime factors and then leave the whole part alone.
The second way: first convert it to an improper fraction, write it into ordinary factors, then reduce the fraction. Convert the already obtained improper fraction into a proper fraction.
Examples can be seen in the photo above.
We really hope that we were able to help you and your children. After all, they are often inattentive in class, so they have to study more intensively at home on their own.
Online calculator performs reduction of algebraic fractions in accordance with the rule of reducing fractions: replacing the original fraction with an equal fraction, but with a smaller numerator and denominator, i.e. Simultaneously dividing the numerator and denominator of a fraction by their common greatest common factor (GCD). The calculator also displays detailed solution, which will help you understand the sequence of the reduction.
Given:
Solution:
Performing fraction reduction
checking the possibility of performing algebraic fraction reduction
1) Determination of the greatest common divisor (GCD) of the numerator and denominator of a fraction
determining the greatest common divisor (GCD) of the numerator and denominator of an algebraic fraction
2) Reducing the numerator and denominator of a fraction
reducing the numerator and denominator of an algebraic fraction
3) Selecting the whole part of a fraction
separating the whole part of an algebraic fraction
4) Converting an algebraic fraction to a decimal fraction
converting an algebraic fraction to decimal
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I. Procedure for reducing an algebraic fraction using an online calculator:
- To reduce an algebraic fraction, enter the values of the numerator and denominator of the fraction in the appropriate fields. If the fraction is mixed, then also fill in the field corresponding to the whole part of the fraction. If the fraction is simple, then leave the whole part field blank.
- To specify a negative fraction, place a minus sign on the whole part of the fraction.
- Depending on the specified algebraic fraction, the following sequence of actions is automatically performed:
- determining the greatest common divisor (GCD) of the numerator and denominator of a fraction;
- reducing the numerator and denominator of a fraction by gcd;
- highlighting the whole part of a fraction, if the numerator of the final fraction is greater than the denominator.
- converting the final algebraic fraction to a decimal fraction rounded to the nearest hundredth.
II. For reference:
A fraction is a number consisting of one or more parts (fractions) of a unit. A common fraction (simple fraction) is written as two numbers (the numerator of the fraction and the denominator of the fraction) separated by a horizontal bar (the fraction bar) indicating the division sign. The numerator of a fraction is the number above the fraction line. The numerator shows how many shares were taken from the whole. The denominator of a fraction is the number below the fraction line. The denominator shows how many equal parts the whole is divided into. A simple fraction is a fraction that does not have a whole part. A simple fraction can be proper or improper. proper fraction - a fraction whose numerator is less than the denominator, so a proper fraction is always less than one. Example of proper fractions: 8/7, 11/19, 16/17. improper fraction - a fraction in which the numerator is greater than or equal to the denominator, therefore an improper fraction is always more than one or equal to it. Example improper fractions: 7/6, 8/7, 13/13. mixed fraction is a number that contains a whole number and a proper fraction, and denotes the sum of that whole number and the proper fraction. Any mixed fraction can be converted to an improper fraction. Example of mixed fractions: 1¼, 2½, 4¾.
III. Note:
- Source data block highlighted yellow , the block of intermediate calculations is highlighted in blue, the solution block is highlighted in green.
- To add, subtract, multiply and divide common or mixed fractions, use the online fraction calculator with detailed solutions.
In this article we will look in detail at how reducing fractions. First, let's discuss what is called reducing a fraction. After this, let's talk about reducing a reducible fraction to an irreducible form. Next we will obtain the rule for reducing fractions and, finally, consider examples of the application of this rule.
Page navigation.
What does it mean to reduce a fraction?
We know that ordinary fractions are divided into reducible and irreducible fractions. From the names you can guess that reducible fractions can be reduced, but irreducible fractions cannot.
What does it mean to reduce a fraction? Reduce a fraction- this means dividing its numerator and denominator by their positive and different from unity. It is clear that as a result of reducing a fraction, a new fraction is obtained with a smaller numerator and denominator, and, due to the basic property of the fraction, the resulting fraction is equal to the original one.
For example, let's reduce the common fraction 8/24 by dividing its numerator and denominator by 2. In other words, let's reduce the fraction 8/24 by 2. Since 8:2=4 and 24:2=12, this reduction results in the fraction 4/12, which is equal to the original fraction 8/24 (see equal and unequal fractions). As a result, we have .
Reducing ordinary fractions to irreducible form
Typically, the ultimate goal of reducing a fraction is to obtain an irreducible fraction that is equal to the original reducible fraction. This goal can be achieved by reducing the original reducible fraction by its numerator and denominator. As a result of such a reduction, an irreducible fraction is always obtained. Indeed, a fraction 
is irreducible, since it is known that 
And 
- . Here we will say that the greatest common divisor of the numerator and denominator of a fraction is the largest number, by which this fraction can be reduced.
So, reducing a common fraction to an irreducible form consists of dividing the numerator and denominator of the original reducible fraction by their gcd.
Let's look at an example, for which we return to the fraction 8/24 and reduce it by the greatest common divisor of the numbers 8 and 24, which is equal to 8. Since 8:8=1 and 24:8=3, we come to the irreducible fraction 1/3. So, .
Note that the phrase “reduce a fraction” often means reducing the original fraction to its irreducible form. In other words, reducing a fraction very often refers to dividing the numerator and denominator by their greatest common factor (rather than by any common factor).
How to reduce a fraction? Rules and examples of reducing fractions
All that remains is to look at the rule for reducing fractions, which explains how to reduce a given fraction.
Rule for reducing fractions consists of two steps:
- firstly, the gcd of the numerator and denominator of the fraction is found;
- secondly, the numerator and denominator of the fraction are divided by their gcd, which gives an irreducible fraction equal to the original one.
Let's sort it out example of reducing a fraction according to the stated rule.
Example.
Reduce the fraction 182/195.
Solution.
Let's carry out both steps prescribed by the rule for reducing a fraction.
First we find GCD(182, 195) . It is most convenient to use the Euclidean algorithm (see): 195=182·1+13, 182=13·14, that is, GCD(182, 195)=13.
Now we divide the numerator and denominator of the fraction 182/195 by 13, and we get the irreducible fraction 14/15, which is equal to the original fraction. This completes the reduction of the fraction.
Briefly, the solution can be written as follows: .
Answer:
This is where we can finish reducing fractions. But to complete the picture, let's look at two more ways to reduce fractions, which are usually used in easy cases.
Sometimes the numerator and denominator of the fraction being reduced is not difficult. Reducing a fraction in this case is very simple: you just need to remove all common factors from the numerator and denominator.
It is worth noting that this method follows directly from the rule of reducing fractions, since the product of all common prime factors of the numerator and denominator is equal to their greatest common divisor.
Let's look at the solution to the example.
Example.
Reduce the fraction 360/2 940.
Solution.
Let's factor the numerator and denominator into simple factors: 360=2·2·2·3·3·5 and 2,940=2·2·3·5·7·7. Thus, 
.
Now we get rid of the common factors in the numerator and denominator; for convenience, we simply cross them out: 
.
Finally, we multiply the remaining factors: , and the reduction of the fraction is completed.
Here is a short summary of the solution: 
.
Answer:
Let's consider another way to reduce a fraction, which consists of sequential reduction. Here, at each step, the fraction is reduced by some common divisor of the numerator and denominator, which is either obvious or easily determined using
