At first glance, algebraic fractions seem very complex, and an untrained student may think that nothing can be done with them. The jumble of variables, numbers and even degrees inspires fear. However, the same rules apply for abbreviating regular (eg 15/25) and algebraic fractions.

Steps

Reducing fractions

Check out the steps for simple fractions. Operations with ordinary and algebraic fractions are similar. For example, let's take the fraction 15/35. To simplify this fraction, one should find common divisor ... Both numbers are divisible by five, so we can highlight 5 in both the numerator and denominator:

15 → 5 * 3 35 → 5 * 7

Now you can reduce common factors, that is, cross out 5 in the numerator and denominator. As a result, we get a simplified fraction 3/7 ... In algebraic expressions, common factors are distinguished in the same way as in ordinary ones. In the previous example, we were able to easily distinguish 5 out of 15 - the same principle applies to more complex expressions such as 15x - 5. Find the common factor. In this case, it will be 5, since both terms (15x and -5) are divisible by 5. As before, select the common factor and carry it over to the left.

15x - 5 = 5 * (3x - 1)

To check if everything is correct, it is enough to multiply the expression in the brackets by 5 - the result will be the same numbers that were at first. Complex members can be selected in the same way as simple ones. For algebraic fractions, the same principles apply as for ordinary ones. This is the easiest way to reduce a fraction. Consider the following fraction:

(x + 2) (x-3)(x + 2) (x + 10)

Note that both the numerator (above) and the denominator (below) contain the term (x + 2), so it can be canceled in the same way as the common factor 5 in the fraction 15/35:

(x + 2) (x-3) → (x-3)(x + 2) (x + 10) → (x + 10)

As a result, we get a simplified expression: (x-3) / (x + 10)

Reduction of algebraic fractions

Find the common factor in the numerator, that is, at the top of the fraction. When canceling an algebraic fraction, the first step is to simplify both of its parts. Start with the numerator and try to expand it into as many factors as possible. Consider the following fraction in this section:

9x-3 15x + 6

Let's start with the numerator: 9x - 3. For 9x and -3, the common factor is 3. Move 3 out of the parentheses, as is done with ordinary numbers: 3 * (3x-1). As a result of this transformation, the following fraction will be obtained:

3 (3x-1) 15x + 6

Find the common factor in the numerator. Let's continue with the above example and write out the denominator: 15x + 6. As before, find the number by which both parts are divisible. And in this case, the common factor is 3, so you can write: 3 * (5x +2). Let's rewrite the fraction as follows:

3 (3x-1) 3 (5x + 2)

Reduce identical members. At this step, you can simplify the fraction. Cancel the identical terms in the numerator and denominator. In our example, this number is 3.

3 (3x-1) → (3x-1) 3 (5x + 2) → (5x + 2)

Determine what the fraction has simplest view... The fraction is completely simplified when there are no common factors left in the numerator and denominator. Note that you cannot cancel out those terms that are inside the parentheses - in the above example, there is no way to separate x from 3x and 5x, since the full terms are (3x -1) and (5x + 2). Thus, the fraction defies further simplification, and the final answer looks like this:

(3x-1)(5x + 2)

Practice cutting fractions yourself. The best way learn the method is independent decision tasks. The correct answers are given below the examples.

4 (x + 2) (x-13)(4x + 8)

Answer:(x = 13)

2x 2 -x 5x

Answer:(2x-1) / 5

Special tricks

Move the negative sign out of the fraction. Suppose the following fraction is given:

3 (x-4) 5 (4-x)

Note that (x-4) and (4-x) are “almost” identical, but they cannot be shortened at once as they are “inverted”. However, (x - 4) can be rewritten as -1 * (4 - x), just as (4 + 2x) can be rewritten as 2 * (2 + x). This is called “reversal of sign”.

-1 * 3 (4-x) 5 (4-x)

Now you can cancel the same terms (4-x):

-1 * 3 (4-x) 5 (4-x)

So, we get the final answer: -3/5 ... Learn to recognize the difference in squares. Difference of squares is when the square of one number is subtracted from the square of another number, as in the expression (a 2 - b 2). The difference of complete squares can always be decomposed into two parts - the sum and the difference of the corresponding square roots... Then the expression will take the following form:

A 2 - b 2 = (a + b) (a-b)

This technique is very useful when looking for common terms in algebraic fractions.

• Check if you have factorized this or that expression correctly. To do this, multiply the factors - the result should be the same expression.
• To completely simplify a fraction, always select the largest factors.

In this lesson we will study the basic property of a fraction, find out which fractions are equal to each other. We will learn how to cancel fractions, determine whether a fraction is cancellable or not, practice reducing fractions and find out when to use cancellation and when not.

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Basic property of a fraction

Imagine this situation.

At the table 3 human and 5 apples. Share 5 apples for three. Each gets a \ (\ mathbf (\ frac (5) (3)) \) apple.

And at the next table still 3 human and also 5 apples. Each one again has \ (\ mathbf (\ frac (5) (3)) \)

Moreover, in total 10 apples 6 human. Each \ (\ mathbf (\ frac (10) (6)) \)

But they are the same thing.

\ (\ mathbf (\ frac (5) (3) = \ frac (10) (6)) \)

These fractions are equivalent.

You can double the number of people and double the number of apples. The result will be the same.

In mathematics, this is formulated as follows:

If the numerator and denominator of a fraction are multiplied or divided by the same number (not equal to 0), then the new fraction will be equal to the original.

This property is sometimes called " the main property of the fraction ».

$$ \ mathbf (\ frac (a) (b) = \ frac (a \ cdot c) (b \ cdot c) = \ frac (a: d) (b: d)) $$

For example, Path from city to village- 14 km.

We walk along the road and determine the distance traveled by kilometer posts. Having passed six columns, six kilometers, we understand that we have passed \ (\ mathbf (\ frac (6) (14)) \) paths.

But if we do not see the columns (maybe they were not installed), we can count the path by electric poles along the road. Their 40 pieces per kilometer. That is, all 560 all the way. Six kilometers- \ (\ mathbf (6 \ cdot40 = 240) \) pillars. That is, we passed 240 from 560 pillars- \ (\ mathbf (\ frac (240) (560)) \)

\ (\ mathbf (\ frac (6) (14) = \ frac (240) (560)) \)

Example 1

Mark the point with coordinates ( 5; 7 ) on coordinate plane XOY... It will match \ (\ mathbf (\ frac (5) (7)) \)

Connect the origin to the resulting point. Construct another point, which has coordinates twice as large as the previous ones. What fraction did you get? Will they be equal?

Solution

A fraction on the coordinate plane can be marked with a point. To represent the fraction \ (\ mathbf (\ frac (5) (7)) \), mark the point with the coordinate 5 along the axis Y and 7 along the axis X... Let's draw a straight line from the origin through our point.

The point corresponding to the fraction \ (\ mathbf (\ frac (10) (14)) \)

They are equivalent: \ (\ mathbf (\ frac (5) (7) = \ frac (10) (14)) \)

Fractions and their abbreviations are another topic that begins in grade 5. Here the base of this action is formed, and then these skills are drawn as a thread into higher mathematics. If the student has not learned, 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 the horizontal line are immediately perceived as a fraction. However, not everyone understands that it can be any number. If it is whole, then it can always be divided by one, then you get an incorrect fraction. But more on that later.

The beginning is always simple. First you need to figure out how to cancel a regular fraction. That is, one in which the numerator is less than the denominator. To do this, you need to remember the basic property of a fraction. It claims that when multiplying (as well as dividing) at the same time its numerator and denominator by the same number, it turns out that it is equivalent to the original fraction.

Division actions that are performed on this property and lead to a reduction. That is, to simplify it as much as possible. The fraction can be reduced as long as there are common factors above and below the line. When they are no longer there, then the reduction is impossible. And they say that this fraction is irreducible.

Two ways

1.Stepwise reduction. It uses a calculation method where both numbers are divided by the minimum common factor that the student has noticed. 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 factor of the numerator and denominator. This is the most rational way to reduce fractions. It involves factoring the numerator and denominator into prime factors. Then you need to choose all the same among them. Their product will give the largest common factor by which the fraction can be canceled.

Both of these methods are equivalent. The student is invited to master them and use the one that he liked best.

What if there are letters and addition and subtraction actions?

With the first part of the question, everything is more or less clear. Letters can be abbreviated in the same way as numbers. The main thing is that they act as multipliers. But with the second, many have problems.

It is important to remember! You can only reduce numbers that are multipliers. 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 master the rule. First, present the numerator and denominator as a product. Then you can cancel if common factors appeared. For representation in the form of multipliers, the following techniques are useful:

• grouping;
• bracketing;
• application of the identities of abbreviated multiplication.

And last way makes it possible to immediately obtain the terms in the form of factors. Therefore, it should always be used if a known pattern is visible.

But this is not scary, then tasks with degrees and roots appear. Then it takes courage to 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 figure out how to cancel fractions with powers, you need to learn two points:

• if there is a sum in the exponent, then it can be decomposed into factors, the degrees of which will be the original terms;
• if the difference, then into the dividend and the divisor, the first will have the decreasing power, the second will have the subtracted.

After completing these steps, the common factors become visible. In such examples, it is not necessary to calculate all degrees. It is enough to simply reduce degrees with the same indicators and bases.

It takes a lot of practice to finally master how to reduce fractions with powers. After several examples of the same type, the 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 about how to reduce a fraction with roots, then you need to divide.

Irrational expressions can also be divided. That is, if the numerator and denominator contain the same factors enclosed under the root sign, then they can be safely reduced. This will simplify the expression and complete the assignment.

If after the reduction under the line of the fraction there is irrationality, then you need to get rid of it. In other words, multiply the numerator and denominator by it. If, after this operation, common factors appear, then they will need to be reduced again.

That's probably all about how to reduce fractions. There are few rules, but there is only one prohibition. Never abbreviate terms!

Let's figure out what cancellation of fractions is, why and how to reduce fractions, give the rule of cancellation of fractions and examples of its use.

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What is fraction reduction

Reduce fraction

To cancel a fraction means to divide its numerator and denominator by a common factor, positive and different from one.

As a result of this action, you get a fraction with a new numerator and denominator, equal to the original fraction.

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

Reducing fractions to an 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 canceled further. Typically, the goal of reducing fractions is to end up with an irreducible fraction. How to bring a fraction to an 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 be mutually prime numbers, and the fraction will be irreducible.

a b = a ÷ NO D (a, b) b ÷ NO D (a, b)

Reducing a fraction to an irreducible form

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

Let's go back to the fraction 6 24 from the first example and bring it to an irreducible form. The greatest common denominator of 6 and 24 is 6. Reduce the fraction:

6 24 = 6 ÷ 6 24 ÷ 6 = 1 4

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

The rule for reducing fractions

To reduce fractions, it is enough to remember the rule, which consists of two steps.

The rule for reducing fractions

To reduce the 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 some practical examples.

Example 1. Reduce the fraction.

The fraction is 182 195. Let's shorten it.

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 OD (182, 195) = 13

Divide the numerator and denominator by 13. We get:

182 195 = 182 ÷ 13 195 ÷ 13 = 14 15

Ready. We got an irreducible fraction, which is equal to the original fraction.

How else can you reduce fractions? In some cases, it is convenient to expand 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

You are given a fraction 360 2940. Let's shorten it.

To do this, we represent 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, as a result of which we get:

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. With this method, the reduction is carried out in several stages, at each of which the fraction is canceled by some obvious common divisor.

Example 3. Reduce the fraction

Reduce the fraction 2000 4400.

You can immediately see that the numerator and denominator have a common factor of 100. Reduce the fraction by 100 and get:

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

20 44 = 20 ÷ 2 44 ÷ 2 = 10 22

Reduce the resulting result by 2 again and get an already irreducible fraction:

10 22 = 10 ÷ 2 22 ÷ 2 = 5 11

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So we got to the reduction. The basic property of a fraction is applied here. BUT! Not so simple. Many fractions (including those from the school course) are quite possible to do with them. And if you take the "cooler" fractions? Let's take a closer look! I recommend looking at materials with fractions.

So, we already know that the numerator and denominator of a fraction can be multiplied and divided by the same number, the fraction will not change from this. Consider three approaches:

The first approach.

For cancellation, divide the numerator and denominator by the common factor. Let's consider some examples:

Let's shorten:

In the examples given, we immediately see which divisors to take for reduction. The process is simple - we iterate over 2,3,4,5 and so on. In most examples of the school course, this is sufficient. But if there is a fraction:

Here the process with the selection of dividers can take a long time;). Of course, such examples lie outside the school course, but you need to be able to cope with them. Below we will see how this is done. For now, let's get back to the reduction process.

As discussed above, in order to reduce the fraction, we carried out division by the common divisor (li) determined by us. That's right! One has only to add signs of divisibility of numbers:

- if the number is even then it is divisible by 2.

- if the number of the last two digits is divisible by 4, then the number itself is divisible by 4.

- if the sum of the digits that make up the number is divisible by 3, then the number itself is divisible by 3. For example 125031, 1 + 2 + 5 + 0 + 3 + 1 = 12. Twelve is divisible by 3, so 123031 is divisible by 3.

- if there is 5 or 0 at the end of the number, then the number is divided by 5.

- if the sum of the digits that make up the number is divisible by 9, then the number itself is divisible by 9. For example, 625032 =.> 6 + 2 + 5 + 0 + 3 + 2 = 18. Eighteen is divisible by 9, so 623032 is divisible by 9.

Second approach.

In a nutshell, in fact, the whole action boils down to factoring the numerator and denominator into factors and then to canceling equal factors in the numerator and denominator (this approach is a consequence of the first approach):

Visually, in order not to get confused and not to be mistaken, equal factors are simply crossed out. The question is - how to factor a number? It is necessary to determine by exhaustive search all the divisors. This is a separate topic, it is not difficult, look at the information in a textbook or on the Internet. You will not encounter any great problems with factoring numbers that are present in fractions of the school course.

Formally, the principle of reduction can be written as follows:

The third approach.

Here is the most interesting for the advanced and those who want to become one. Reduce the fraction 143/273. Try it yourself! So how did it work out quickly? Now look!

We turn it over (we swap the numerator and denominator). Divide the resulting fraction with a corner and convert it to a mixed number, that is, select the whole part:

It's already easier. We can see that the numerator and denominator can be canceled by 13:

And now do not forget to turn the fraction back again, let's write down the whole chain:

Checked - it takes less time than searching and checking the divisors. Let's go back to our two examples:

First. Divide with a corner (not on a calculator), we get:

This fraction is simpler, of course, but there is again a problem with the reduction. Now we separately parse the fraction 1273/1463, turn it over:

It's already easier here. We can consider such a divisor as 19. The rest do not fit, it can be seen: 190: 19 = 10, 1273: 19 = 67. Hurray! Let's write down:

Next example. Let's shorten 88179/2717.

Divide, we get:

Separately we parse the fraction 1235/2717, turn it over:

We can consider such a divisor as 13 (up to 13 are not suitable):

Numerator 247: 13 = 19 Denominator 1235: 13 = 95

* In the process, we saw another divisor equal to 19. It turns out that:

Now we write down the original number:

And it doesn't matter what will be more in the fraction - the numerator or the denominator, if the denominator, then we turn it over and act as described. Thus, we can reduce any fraction, the third approach can be called universal.

Of course, the two examples discussed above are not easy examples. Let's try this technology on the "simple" fractions we have already considered:

Two fourths.

Seventy two sixties. The numerator is greater than the denominator, you do not need to turn it over:

Of course, the third approach was applied to such simple examples just as an alternative. The method, as already mentioned, is universal, but not convenient and correct for all fractions, this especially applies to simple ones.

The variety of fractions is great. It is important that you learn exactly the principles. There is simply no strict rule for working with fractions. We looked, figured out how it is more convenient to act and go ahead. With practice, you will get the skill and you will click them like seeds.

Output:

If you see common (s) divisor (s) for the numerator and denominator, then use them to reduce.

If you know how to quickly factor a number, then expand the numerator and denominator, then reduce.

If you cannot determine the common divisor in any way, then use the third approach.

* To reduce fractions, it is important to learn the principles of reduction, to understand the basic property of a fraction, to know the approaches to the solution, to be extremely careful in calculations.

And remember! It is customary to reduce a fraction to the stop, that is, to reduce it while there is a common divisor.

Best regards, Alexander Krutitskikh.