Without knowing how to reduce a fraction and having a stable skill in solving such examples, it is very difficult to study algebra in school. The further you go, the more it interferes with your basic knowledge of reducing fractions. new information. First, powers appear, then factors, which later become polynomials.
How can you avoid getting confused here? Thoroughly consolidate skills in previous topics and gradually prepare for knowledge of how to reduce a fraction, which becomes more complex from year to year.
Basic knowledge
Without them, you will not be able to cope with tasks of any level. To understand, you need to understand two simple points. First: you can only reduce factors. This nuance turns out to be very important when polynomials appear in the numerator or denominator. Then you need to clearly distinguish where the multiplier is and where the addend is.
The second point says that any number can be represented in the form of factors. Moreover, the result of reduction is a fraction whose numerator and denominator can no longer be reduced.
Rules for reducing common fractions
First, you should check whether the numerator is divisible by the denominator or vice versa. Then it is precisely this number that needs to be reduced. This is the simplest option.
The second is analysis appearance numbers. If both end in one or more zeros, then they can be shortened by 10, 100 or a thousand. Here you can notice whether the numbers are even. If yes, then you can safely cut it by two.
The third rule for reducing a fraction is to factor the numerator and denominator into prime factors. At this time, you need to actively use all your knowledge about the signs of divisibility of numbers. After this decomposition, all that remains is to find all the repeating ones, multiply them and reduce them by the resulting number.
What if there is an algebraic expression in a fraction?
This is where the first difficulties appear. Because this is where terms appear that can be identical to factors. I really want to reduce them, but I can’t. Before you can reduce an algebraic fraction, it must be converted so that it has factors.
To do this, you will need to perform several steps. You may need to go through all of them, or maybe the first one will provide a suitable option.
Check whether the numerator and denominator or any expression in them differ by sign. In this case, you just need to put minus one out of brackets. This produces equal factors that can be reduced.
See if it is possible to remove the common factor from the polynomial out of brackets. Perhaps this will result in a parenthesis, which can also be shortened, or it will be a removed monomial.
Try to group the monomials in order to then add a common factor to them. After this, it may turn out that there will be factors that can be reduced, or again the bracketing of common elements will be repeated.
Try to consider abbreviated multiplication formulas in writing. With their help, you can easily convert polynomials into factors.
Sequence of operations with fractions with powers
In order to easily understand the question of how to reduce a fraction with powers, you need to firmly remember the basic operations with them. The first of these is related to the multiplication of powers. In this case, if the bases are the same, the indicators must be added.
The second is division. Again, for those that have the same reasons, the indicators will need to be subtracted. Moreover, you need to subtract from the number that is in the dividend, and not vice versa.
The third is exponentiation. In this situation, the indicators are multiplied.
Successful reduction will also require the ability to reduce powers to equal bases. That is, to see that four is two squared. Or 27 - the cube of three. Because reducing 9 squared and 3 cubed is difficult. But if we transform the first expression as (3 2) 2, then the reduction will be successful.
In this lesson we will study the basic property of a fraction, find out which fractions are equal to each other. We'll learn to reduce fractions, determine whether a fraction is reducible or not, practice reducing fractions, and learn when to use a contraction and when not to.
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The main property of a fraction
Imagine this situation.
At the table 3 person and 5 apples Share 5 apples for three. Everyone gets \(\mathbf(\frac(5)(3))\) apples.
And at the next table 3 person and too 5 apples Each again \(\mathbf(\frac(5)(3))\)
In total 10 apples 6 Human. Each \(\mathbf(\frac(10)(6))\)
But it's 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 it is formulated like this:
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 " main property of a fraction ».
$$\mathbf(\frac(a)(b) = \frac(a\cdot c)(b\cdot c) = \frac(a:d)(b:d))$$
For example, The path from city to village - 14 km.
We walk along the road and determine the distance traveled by kilometer markers. Having walked six columns, six kilometers, we understand that we have covered \(\mathbf(\frac(6)(14))\) distance.
But if we don’t see the poles (maybe they weren’t installed), we can calculate the path using the electric poles along the road. Their 40 pieces for every kilometer. That is, in total 560 all the way. Six kilometers - \(\mathbf(6\cdot40 = 240)\) pillars. That is, we have passed 240 from 560 pillars-\(\mathbf(\frac(240)(560))\)
\(\mathbf(\frac(6)(14) = \frac(240)(560))\)
Example 1
Mark a point with coordinates ( 5; 7 ) on the coordinate plane XOY. It will correspond to the fraction \(\mathbf(\frac(5)(7))\)
Connect the origin of coordinates to the resulting point. Construct another point that has coordinates twice the previous ones. What fraction did you get? Will they be equal?
Solution
A fraction on the coordinate plane can be marked with a dot. 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 \(\mathbf(\frac(10)(14))\) will also lie on the same line
They are equivalent: \(\mathbf(\frac(5)(7) = \frac(10)(14))\)
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 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 factors in the numerator and denominator 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 By given type It is quite difficult for a fraction 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 for 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 about 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 ;
Because the common fractions are a special case of algebraic fractions, let us recall how their reduction is carried out. 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'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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So we got to the reduction. The basic property of a fraction is applied here. BUT! Not so simple. With many fractions (including those from the school course), it is quite possible to get by with them. What if we take fractions that are “more abrupt”? 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. Let's consider three approaches:
Approach one.
To reduce, divide the numerator and denominator by common divisor. Let's look at examples:
Let's shorten:
In the examples given, we immediately see which divisors to take for reduction. The process is simple - we go through 2,3,4,5 and so on. In most school course examples, this is quite enough. But if it’s a fraction:
Here the process of selecting divisors can take a long time;). Of course, such examples are outside the school curriculum, but you need to be able to cope with them. Below we will look at how this is done. For now, let's get back to the downsizing process.
As discussed above, in order to reduce a fraction, we divided by the common divisor(s) we determined. Everything is correct! One has only to add signs of divisibility of numbers:
- if the number is even, then it is divisible by 2.
- if a number from 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 the number ends with 5 or 0, then the number is divisible 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, which means 623032 is divisible by 9.
Second approach.
To put it briefly, in fact, the whole action comes down to factoring the numerator and denominator and then reducing equal factors in the numerator and denominator (this approach is a consequence of the first approach):
Visually, in order to avoid confusion and mistakes, equal factors are simply crossed out. Question - how to factor a number? It is necessary to determine all divisors by searching. This is a separate topic, it is not complicated, look up the information in a textbook or on the Internet. You won't encounter any big problems with factoring numbers that are present in school fractions.
Formally, the reduction principle can be written as follows:
Approach three.
Here is the most interesting thing for the advanced and those who want to become one. Let's reduce the fraction 143/273. Try it yourself! Well, how did it happen quickly? Now look!
We turn it over (we change places of the numerator and denominator). We divide the resulting fraction with a corner and convert it into a mixed number, that is, we select the whole part:
It's already easier. We see that the numerator and denominator can be reduced by 13:
Now don’t forget to flip the fraction back again, let’s write down the whole chain:
Checked - it takes less time than searching through and checking divisors. Let's return to our two examples:
First. Divide with a corner (not on a calculator), we get:
This fraction is simpler, of course, but the reduction is again a problem. Now we separately analyze the fraction 1273/1463 and turn it over:
It's easier here. We can consider a divisor such as 19. The rest are not suitable, this is clear: 190:19 = 10, 1273:19 = 67. Hurray! Let's write down:
Next example. Let's shorten 88179/2717.
Divide, we get:
Separately, we analyze the fraction 1235/2717 and turn it over:
We can consider a divisor such as 13 (up to 13 is not suitable):
Numerator 247:13=19 Denominator 1235:13=95
*During 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 is larger in the fraction - the numerator or the denominator, if it is the denominator, then we turn it over and act as described. This way we can reduce any fraction; the third approach can be called universal.
Of course, the two examples discussed above are not simple examples. Let's try this technology on the “simple” fractions we have already considered:
Two quarters.
Seventy-two sixties. The numerator is greater than the denominator; there is no need to reverse it:
Of course, the third approach was applied to such simple examples simply as an alternative. The method, as already said, is universal, but not convenient and correct for all fractions, especially for simple ones.
The variety of fractions is great. It is important that you understand the principles. There is simply no strict rule for working with fractions. We looked, figured out how it would be more convenient to act, and moved forward. With practice, skill will come and you will crack them like seeds.
Conclusion:
If you see a common divisor(s) for the numerator and denominator, use them to reduce.
If you know how to quickly factor a number, then factor the numerator and denominator, then reduce.
If you can’t determine the common divisor, then use the third approach.
*To reduce fractions, it is important to master the principles of reduction, understand the basic property of a fraction, know approaches to solving, and be extremely careful when making calculations.
And remember! It is customary to reduce a fraction until it stops, that is, reduce it as long as there is a common divisor.
Sincerely, Alexander Krutitskikh.
It is based on their basic property: if the numerator and denominator of a fraction are divided by the same non-zero polynomial, then an equal fraction will be obtained.
You can only reduce multipliers!
Members of polynomials cannot be abbreviated!
To reduce an algebraic fraction, the polynomials in the numerator and denominator must first be factorized.
Let's look at examples of reducing fractions.
The numerator and denominator of the fraction contain monomials. They represent work(numbers, variables and their powers), multipliers we can reduce.
We reduce the numbers by their greatest common divisor, that is, by greatest number, by which each of these numbers is divided. For 24 and 36 this is 12. After reduction, 2 remains from 24, and 3 from 36.
We reduce the degrees by the degree with the lowest index. To reduce a fraction means to divide the numerator and denominator by the same divisor, and subtract the exponents.
a² and a⁷ are reduced to a². In this case, one remains in the numerator of a² (we write 1 only in the case when, after reduction, there are no other factors left. From 24, 2 remains, so we do not write 1 remaining from a²). From a⁷, after reduction, a⁵ remains.
b and b are reduced by b; the resulting units are not written.
c³º and c⁵ are shortened to c⁵. From c³º what remains is c²⁵, from c⁵ is one (we don’t write it). Thus,
The numerator and denominator of this algebraic fraction are polynomials. You cannot cancel terms of polynomials! (you cannot reduce, for example, 8x² and 2x!). To reduce this fraction, you need . The numerator has a common factor of 4x. Let's take it out of brackets:
Both the numerator and denominator have the same factor (2x-3). We reduce the fraction by this factor. In the numerator we got 4x, in the denominator - 1. According to 1 property of algebraic fractions, the fraction is equal to 4x.
You can only reduce factors (you cannot reduce this fraction by 25x²!). Therefore, the polynomials in the numerator and denominator of the fraction must be factorized.
The numerator is the complete square of the sum, the denominator is the difference of squares. After decomposition using abbreviated multiplication formulas, we obtain:
We reduce the fraction by (5x+1) (to do this, cross out the two in the numerator as an exponent, leaving (5x+1)² (5x+1)):
The numerator has a common factor of 2, let's take it out of brackets. The denominator is the formula for the difference of cubes:
As a result of the expansion, the numerator and denominator received the same factor (9+3a+a²). We reduce the fraction by it:
The polynomial in the numerator consists of 4 terms. the first term with the second, the third with the fourth, and remove the common factor x² from the first brackets. We decompose the denominator using the sum of cubes formula:
In the numerator, let’s take the common factor (x+2) out of brackets:
Reduce the fraction by (x+2):
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