Let's understand what reducing fractions is, why and how to reduce fractions, and give the rule for reducing fractions and examples of its use.

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What is "reducing fractions"

To reduce a fraction is to divide its numerator and denominator by a common factor that is positive and different from one.

As a result of this action, a fraction with a new numerator and denominator will be obtained, equal to the original fraction.

For example, let's take the common fraction 6 24 and reduce it. Divide the numerator and denominator by 2, resulting in 6 24 = 6 ÷ 2 24 ÷ 2 = 3 12. In this example, we reduced the original fraction by 2.

Reducing fractions to irreducible form

In the previous example, we reduced the fraction 6 24 by 2, resulting in the fraction 3 12. It is easy to see that this fraction can be further reduced. Typically, the goal of reducing fractions is to end up with an irreducible fraction. How to reduce a fraction to its irreducible form?

This can be done by reducing the numerator and denominator by their greatest common factor (GCD). Then, by the property of the greatest common divisor, the numerator and denominator will have mutually prime numbers, and the fraction will be irreducible.

a b = a ÷ N O D (a , b) b ÷ N O D (a , b)

Reducing a fraction to an irreducible form

To reduce a fraction to an irreducible form, you need to divide its numerator and denominator by their gcd.

Let's return to the fraction 6 24 from the first example and bring it to its irreducible form. The greatest common divisor of the numbers 6 and 24 is 6. Let's reduce the fraction:

6 24 = 6 ÷ 6 24 ÷ 6 = 1 4

Reducing fractions is convenient to use so as not to work with large numbers. In general, there is an unspoken rule in mathematics: if you can simplify any expression, then you need to do it. Reducing a fraction most often means reducing it to an irreducible form, and not simply reducing it by the common divisor of the numerator and denominator.

Rule for reducing fractions

To reduce fractions, just remember the rule, which consists of two steps.

Rule for reducing fractions

To reduce a fraction you need:

1. Find the gcd of the numerator and denominator.
2. Divide the numerator and denominator by their gcd.

Let's look at practical examples.

Example 1. Let's reduce the fraction.

Given the fraction 182 195. Let's shorten it.

Let's find the gcd of the numerator and denominator. To do this, in this case it is most convenient to use the Euclidean algorithm.

195 = 182 1 + 13 182 = 13 14 N O D (182, 195) = 13

Divide the numerator and denominator by 13. We get:

182 195 = 182 ÷ 13 195 ÷ 13 = 14 15

Ready. We have obtained an irreducible fraction that is equal to the original fraction.

How else can you reduce fractions? In some cases, it is convenient to factor the numerator and denominator into prime factors, and then remove all common factors from the upper and lower parts of the fraction.

Example 2. Reduce the fraction

Given the fraction 360 2940. Let's shorten it.

To do this, imagine the original fraction in the form:

360 2940 = 2 2 2 3 3 5 2 2 3 5 7 7

Let's get rid of the common factors in the numerator and denominator, resulting in:

360 2940 = 2 2 2 3 3 5 2 2 3 5 7 7 = 2 3 7 7 = 6 49

Finally, let's look at another way to reduce fractions. This is the so-called sequential reduction. Using this method, the reduction is carried out in several stages, in each of which the fraction is reduced by some obvious common factor.

Example 3. Reduce the fraction

Let's reduce the fraction 2000 4400.

It is immediately clear that the numerator and denominator have a common factor of 100. We reduce the fraction by 100 and get:

2000 4400 = 2000 ÷ 100 4400 ÷ 100 = 20 44

20 44 = 20 ÷ 2 44 ÷ 2 = 10 22

We reduce the resulting result again by 2 and obtain an irreducible fraction:

10 22 = 10 ÷ 2 22 ÷ 2 = 5 11

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Fractions

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

Fractions are not much of a nuisance in high school. For the time being. Until you come across powers with rational exponents and logarithms. And there... You press and press the calculator, and it shows a full display of some numbers. You have to think with your head like in the third grade.

Let's finally figure out fractions! Well, how much can you get confused in them!? Moreover, it’s all simple and logical. So, what are the types of fractions?

Types of fractions. Transformations.

There are three types of fractions.

1. Common fractions , For example:

Sometimes instead of a horizontal line they put a slash: 1/2, 3/4, 19/5, well, and so on. Here we will often use this spelling. The top number is called numerator, lower - denominator. If you constantly confuse these names (it happens...), say to yourself the phrase: " Zzzzz remember! Zzzzz denominator - look zzzzz uh!" Look, everything will be zzz remembered.)

The dash, either horizontal or inclined, means division the top number (numerator) to the bottom (denominator). That's all! Instead of a dash, it is quite possible to put a division sign - two dots.

When complete division is possible, this must be done. So, instead of the fraction “32/8” it is much more pleasant to write the number “4”. Those. 32 is simply divided by 8.

32/8 = 32: 8 = 4

I'm not even talking about the fraction "4/1". Which is also just "4". And if it’s not completely divisible, we leave it as a fraction. Sometimes you have to do the opposite operation. Convert a whole number into a fraction. But more on that later.

2. Decimals , For example:

It is in this form that you will need to write down the answers to tasks “B”.

3. Mixed numbers , For example:

Mixed numbers are practically not used in high school. In order to work with them, they must be converted into ordinary fractions. But you definitely need to be able to do this! Otherwise you will come across such a number in a problem and freeze... Out of nowhere. But we will remember this procedure! A little lower.

Most versatile common fractions. Let's start with them. By the way, if a fraction contains all sorts of logarithms, sines and other letters, this does not change anything. In the sense that everything actions with fractional expressions are no different from actions with ordinary fractions!

The main property of a fraction.

So, let's go! To begin with, I will surprise you. The whole variety of fraction transformations is provided by one single property! That's what it's called main property of a fraction. Remember: If the numerator and denominator of a fraction are multiplied (divided) by the same number, the fraction does not change. Those:

It is clear that you can continue to write until you are blue in the face. Don’t let sines and logarithms confuse you, we’ll deal with them further. The main thing is to understand that all these various expressions are the same fraction . 2/3.

Do we need it, all these transformations? Yes! Now you will see for yourself. To begin with, let's use the basic property of a fraction for reducing fractions. It would seem like an elementary thing. Divide the numerator and denominator by the same number and that's it! It's impossible to make a mistake! But... man is a creative being. You can make a mistake anywhere! Especially if you have to reduce not a fraction like 5/10, but a fractional expression with all sorts of letters.

How to correctly and quickly reduce fractions without doing extra work can be read in the special Section 555.

A normal student doesn't bother dividing the numerator and denominator by the same number (or expression)! He simply crosses out everything that is the same above and below! This is where a typical mistake, a blunder, if you will, lurks.

For example, you need to simplify the expression:

There’s nothing to think about here, cross out the letter “a” on top and the “2” on the bottom! We get:

Everything is correct. But really you divided all numerator and all the denominator is "a". If you are used to just crossing out, then in a hurry you can cross out the “a” in the expression

and get it again

Which would be categorically untrue. Because here all the numerator on "a" is already not shared! This fraction cannot be reduced. By the way, such a reduction is, um... a serious challenge for the teacher. This is not forgiven! Do you remember? When reducing, you need to divide all numerator and all denominator!

Reducing fractions makes life a lot easier. You will get a fraction somewhere, for example 375/1000. How can I continue to work with her now? Without a calculator? Multiply, say, add, square!? And if you’re not too lazy, and carefully cut it down by five, and by another five, and even... while it’s being shortened, in short. Let's get 3/8! Much nicer, right?

The main property of a fraction allows you to convert ordinary fractions to decimals and vice versa without a calculator! This is important for the Unified State Exam, right?

How to convert fractions from one type to another.

With decimal fractions everything is simple. As it is heard, so it is written! Let's say 0.25. This is zero point twenty five hundredths. So we write: 25/100. We reduce (we divide the numerator and denominator by 25), we get the usual fraction: 1/4. All. It happens, and nothing is reduced. Like 0.3. This is three tenths, i.e. 3/10.

What if the integers are not zero? It's OK. We write down the whole fraction without any commas in the numerator, and in the denominator - what is heard. For example: 3.17. This is three point seventeen hundredths. We write 317 in the numerator and 100 in the denominator. We get 317/100. Nothing is reduced, that means everything. This is the answer. Elementary, Watson! From all that has been said, a useful conclusion: any decimal fraction can be converted to a common fraction .

But some people cannot do the reverse conversion from ordinary to decimal without a calculator. And it is necessary! How will you write down the answer on the Unified State Exam!? Read carefully and master this process.

What is the characteristic of a decimal fraction? Her denominator is Always costs 10, or 100, or 1000, or 10000 and so on. If your common fraction has a denominator like this, there's no problem. For example, 4/10 = 0.4. Or 7/100 = 0.07. Or 12/10 = 1.2. What if the answer to the task in section “B” turned out to be 1/2? What will we write in response? Decimals are required...

Let's remember main property of a fraction ! Mathematics favorably allows you to multiply the numerator and denominator by the same number. Anything, by the way! Except zero, of course. So let’s use this property to our advantage! What can the denominator be multiplied by, i.e. 2 so that it becomes 10, or 100, or 1000 (smaller is better, of course...)? At 5, obviously. Feel free to multiply the denominator (this is us necessary) by 5. But then the numerator must also be multiplied by 5. This is already mathematics demands! We get 1/2 = 1x5/2x5 = 5/10 = 0.5. That's it.

However, all sorts of denominators come across. You will come across, for example, the fraction 3/16. Try and figure out what to multiply 16 by to make 100, or 1000... Doesn’t it work? Then you can simply divide 3 by 16. In the absence of a calculator, you will have to divide with a corner, on a piece of paper, as they taught in elementary school. We get 0.1875.

And there are also very bad denominators. For example, there is no way to turn the fraction 1/3 into a good decimal. Both on the calculator and on a piece of paper, we get 0.3333333... This means that 1/3 is an exact decimal fraction not translated. Same as 1/7, 5/6 and so on. There are many of them, untranslatable. This brings us to another useful conclusion. Not every fraction can be converted to a decimal !

By the way, this is useful information for self-testing. In section "B" you must write down a decimal fraction in your answer. And you got, for example, 4/3. This fraction does not convert to a decimal. This means you made a mistake somewhere along the way! Go back and check the solution.

So, we figured out ordinary and decimal fractions. All that remains is to deal with mixed numbers. To work with them, they must be converted into ordinary fractions. How to do this? You can catch a sixth grader and ask him. But a sixth grader won’t always be at hand... You’ll have to do it yourself. It's not difficult. You need to multiply the denominator of the fractional part by the whole part and add the numerator of the fractional part. This will be the numerator of the common fraction. What about the denominator? The denominator will remain the same. It sounds complicated, but in reality everything is simple. Let's look at an example.

Suppose you were horrified to see the number in the problem:

Calmly, without panic, we think. The whole part is 1. Unit. The fractional part is 3/7. Therefore, the denominator of the fractional part is 7. This denominator will be the denominator of the ordinary fraction. We count the numerator. We multiply 7 by 1 (the integer part) and add 3 (the numerator of the fractional part). We get 10. This will be the numerator of a common fraction. That's it. It looks even simpler in mathematical notation:

Is it clear? Then secure your success! Convert to ordinary fractions. You should get 10/7, 7/2, 23/10 and 21/4.

The reverse operation - converting an improper fraction to a mixed number - is rarely required in high school. Well, if so... And if you are not in high school, you can look into the special Section 555. By the way, you will also learn about improper fractions there.

Well, that's practically all. You remembered the types of fractions and understood How transfer them from one type to another. The question remains: For what do this? Where and when to apply this deep knowledge?

I answer. Any example itself suggests the necessary actions. If in the example ordinary fractions, decimals, and even mixed numbers are mixed together, we convert everything into ordinary fractions. It can always be done. Well, if it says something like 0.8 + 0.3, then we count it that way, without any translation. Why do we need extra work? We choose the solution that is convenient us !

If the task is all decimal fractions, but um... some kind of evil ones, go to ordinary ones and try it! Look, everything will work out. For example, you will have to square the number 0.125. It’s not so easy if you haven’t gotten used to using a calculator! Not only do you have to multiply numbers in a column, you also have to think about where to insert the comma! It definitely won’t work in your head! What if we move on to an ordinary fraction?

0.125 = 125/1000. We reduce it by 5 (this is for starters). We get 25/200. Once again by 5. We get 5/40. Oh, it's still shrinking! Back to 5! We get 1/8. We can easily square it (in our minds!) and get 1/64. All!

Let's summarize this lesson.

1. There are three types of fractions. Common, decimal and mixed numbers.

2. Decimals and mixed numbers Always can be converted to ordinary fractions. Reverse transfer not always possible

3. The choice of the type of fractions to work with a task depends on the task itself. If there are different types of fractions in one task, the most reliable thing is to switch to ordinary fractions.

Now you can practice. First, convert these decimal fractions to ordinary fractions:

3,8; 0,75; 0,15; 1,4; 0,725; 0,012

You should get answers like this (in a mess!):

Let's finish here. In this lesson we refreshed our memory on key points about fractions. It happens, however, that there is nothing special to refresh...) If someone has completely forgotten, or has not yet mastered it... Then you can go to a special Section 555. All the basics are covered in detail there. Many suddenly understand everything are starting. And they solve fractions on the fly).

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

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 a 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. In simple terms, 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.

1. The first rule of reducing fractions is to first find the greatest common factor of the denominator and numerator of your fraction.
2. 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 one.

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.

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 whose numerator is less than its 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 in this property and result in 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, represent 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 contain 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 prohibition. Never shorten terms!

This article continues the topic of converting 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 of a simpler form, at best 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 quite 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 decomposed 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 some 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 general algebraic fractions, in which the numerators and denominators can be either monomials or polynomials, there can be a problem where 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's factor the polynomials in the numerator and denominator. Let's put 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 have a 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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