Online calculator. Calculating expressions with numerical fractions. Multiplying, subtracting, dividing, adding and reducing fractions with different denominators

Actions with fractions. In this article we will look at examples, everything in detail with explanations. We will consider common fractions. We'll look at decimals later. I recommend watching the whole thing and studying it sequentially.

1. Sum of fractions, difference of fractions.

Rule: when adding fractions with equal denominators, the result is a fraction - the denominator of which remains the same, and its numerator will be equal to the sum of the numerators of the fractions.

Rule: when calculating the difference between fractions with the same denominators, we obtain a fraction - the denominator remains the same, and the numerator of the second is subtracted from the numerator of the first fraction.

Formal notation for the sum and difference of fractions with equal denominators:

Examples (1):

It is clear that when ordinary fractions are given, then everything is simple, but what if they are mixed? Nothing complicated...

Option 1– you can convert them into ordinary ones and then calculate them.

Option 2– you can “work” separately with the integer and fractional parts.

Examples (2):

More:

And if the difference of two is given mixed fractions and the numerator of the first fraction will be less than the numerator of the second? You can also act in two ways.

Examples (3):

*Converted to ordinary fractions, calculated the difference, converted the resulting improper fraction to a mixed fraction.

*We broke it down into integer and fractional parts, got a three, then presented 3 as the sum of 2 and 1, with one represented as 11/11, then found the difference between 11/11 and 7/11 and calculated the result. The meaning of the above transformations is to take (select) a unit and present it in the form of a fraction with the denominator we need, then we can subtract another from this fraction.

Another example:

Conclusion: there is a universal approach - in order to calculate the sum (difference) of mixed fractions with equal denominators, they can always be converted to improper ones, then perform required action. After this, if the result is an improper fraction, we convert it to a mixed fraction.

Above we looked at examples with fractions that have equal denominators. What if the denominators are different? In this case, the fractions are reduced to the same denominator and the specified action is performed. To change (transform) a fraction, the basic property of the fraction is used.

Let's look at simple examples:

In these examples, we immediately see how one of the fractions can be transformed to get equal denominators.

If we designate ways to reduce fractions to the same denominator, then we will call this one METHOD ONE.

That is, immediately when “estimating” a fraction, you need to figure out whether this approach will work - we check whether the larger denominator is divisible by the smaller one. And if it is divisible, then we carry out the transformation - we multiply the numerator and denominator so that the denominators of both fractions become equal.

Now look at these examples:

This approach is not applicable to them. There are also ways to reduce fractions to a common denominator; let’s consider them.

Method TWO.

We multiply the numerator and denominator of the first fraction by the denominator of the second, and the numerator and denominator of the second fraction by the denominator of the first:

*In fact, we reduce fractions to the form when the denominators become equal. Next, we use the rule for adding fractions with equal denominators.

Example:

*This method can be called universal, and it always works. The only negative is that after the calculations you may end up with a fraction that will need to be further reduced.

Let's look at an example:

It can be seen that the numerator and denominator are divisible by 5:

Method THREE.

You need to find the least common multiple (LCM) of the denominators. This will be the common denominator. What kind of number is this? This is the smallest natural number that is divisible by each of the numbers.

Look, here are two numbers: 3 and 4, there are many numbers that are divisible by them - these are 12, 24, 36, ... The smallest of them is 12. Or 6 and 15, they are divisible by 30, 60, 90 .... The least is 30. The question is - how to determine this least common multiple?

There is a clear algorithm, but often this can be done immediately without calculations. For example, according to the above examples (3 and 4, 6 and 15) no algorithm is needed, we took large numbers (4 and 15), doubled them and saw that they are divisible by the second number, but pairs of numbers can be others, for example 51 and 119.

Algorithm. In order to determine the least common multiple of several numbers, you must:

- decompose each number into SIMPLE factors

— write down the decomposition of the BIGGER of them

- multiply it by the MISSING factors of other numbers

Let's look at examples:

50 and 60 => 50 = 2∙5∙5 60 = 2∙2∙3∙5

in decomposition more one five is missing

=> LCM(50,60) = 2∙2∙3∙5∙5 = 300

48 and 72 => 48 = 2∙2∙2∙2∙3 72 = 2∙2∙2∙3∙3

in the expansion of a larger number two and three are missing

=> LCM(48.72) = 2∙2∙2∙2∙3∙3 = 144

* Least common multiple of two prime numbers equal to their product

Question! Why is finding the least common multiple useful, since you can use the second method and simply reduce the resulting fraction? Yes, it is possible, but it is not always convenient. Look at the denominator for the numbers 48 and 72 if you simply multiply them 48∙72 = 3456. You will agree that it is more pleasant to work with smaller numbers.

Let's look at examples:

*51 = 3∙17 119 = 7∙17

the expansion of a larger number is missing a triple

=> NOC(51,119) = 3∙7∙17

Now let's use the first method:

*Look at the difference in the calculations, in the first case there are a minimum of them, but in the second you need to work separately on a piece of paper, and even the fraction you received needs to be reduced. Finding the LOC simplifies the work significantly.

More examples:

*In the second example it is clear that smallest number which is divisible by 40 and 60 is equal to 120.

RESULT! GENERAL COMPUTING ALGORITHM!

— we reduce fractions to ordinary ones if there is an integer part.

- we bring fractions to a common denominator (first we look at whether one denominator is divisible by another; if it is divisible, then we multiply the numerator and denominator of this other fraction; if it is not divisible, we act using the other methods indicated above).

- Having received fractions with equal denominators, we perform operations (addition, subtraction).

- if necessary, we reduce the result.

- if necessary, then select the whole part.

2. Product of fractions.

The rule is simple. When multiplying fractions, their numerators and denominators are multiplied:

Examples:

Task. 13 tons of vegetables were brought to the base. Potatoes make up ¾ of all imported vegetables. How many kilograms of potatoes were brought to the base?

Let's finish with the piece.

*I previously promised to give you a formal explanation of the main property of a fraction through a product, please:

3. Division of fractions.

Dividing fractions comes down to multiplying them. It is important to remember here that the fraction that is the divisor (the one that is divided by) is turned over and the action changes to multiplication:

This action can be written in the form of a so-called four-story fraction, because the division “:” itself can also be written as a fraction:

Examples:

That's all! Good luck to you!

Sincerely, Alexander Krutitskikh.

Now that we have learned how to add and multiply individual fractions, we can look at more complex designs. For example, what if the same problem involves adding, subtracting, and multiplying fractions?

First of all, you need to convert all fractions to improper ones. Then we perform the required actions sequentially - in the same order as for ordinary numbers. Namely:

1. Exponentiation is done first - get rid of all expressions containing exponents;
2. Then - division and multiplication;
3. The last step is addition and subtraction.

Of course, if there are parentheses in the expression, the order of operations changes - everything that is inside the parentheses must be counted first. And remember about improper fractions: you need to highlight the whole part only when all other actions have already been completed.

Let's convert all the fractions from the first expression to improper ones, and then perform the following steps:

Now let's find the value of the second expression. Here fractions with whole part no, but there are parentheses, so we do the addition first, and only then the division. Note that 14 = 7 · 2. Then:

Finally, consider the third example. There are brackets and a degree here - it is better to count them separately. Considering that 9 = 3 3, we have:

Pay attention to the last example. To raise a fraction to a power, you must separately raise the numerator to this power, and separately, the denominator.

You can decide differently. If we recall the definition of degree, the problem reduces to ordinary multiplication fractions:

Multistory fractions

Until now, we have considered only “pure” fractions, when the numerator and denominator are ordinary numbers. This is quite consistent with the definition of a number fraction given in the very first lesson.

But what if you put a more complex object in the numerator or denominator? For example, another numerical fraction? Such constructions arise quite often, especially when working with long expressions. Here are a couple of examples:

There is only one rule for working with multi-level fractions: you must get rid of them immediately. Removing “extra” floors is quite simple, if you remember that the slash means the standard division operation. Therefore, any fraction can be rewritten as follows:

Using this fact and following the procedure, we can easily reduce any multi-story fraction to an ordinary one. Take a look at the examples:

Task. Convert multistory fractions to ordinary ones:

In each case, we rewrite the main fraction, replacing the dividing line with a division sign. Also remember that any integer can be represented as a fraction with a denominator of 1. That is 12 = 12/1; 3 = 3/1. We get:

In the last example, the fractions were canceled before the final multiplication.

Specifics of working with multi-level fractions

There is one subtlety in multi-level fractions that must always be remembered, otherwise you can get the wrong answer, even if all the calculations were correct. Take a look:

1. The numerator contains the single number 7, and the denominator contains the fraction 12/5;
2. The numerator contains the fraction 7/12, and the denominator contains the separate number 5.

So, for one recording we got two completely different interpretations. If you count, the answers will also be different:

To ensure that the record is always read unambiguously, use a simple rule: the dividing line of the main fraction must be longer than the line of the nested fraction. Preferably several times.

If you follow this rule, then the above fractions should be written as follows:

Yes, it's probably unsightly and takes up too much space. But you will count correctly. Finally, a couple of examples where multi-story fractions actually arise:

Task. Find the meanings of the expressions:

So, let's work with the first example. Let's convert all fractions to improper ones, and then perform addition and division operations:

Let's do the same with the second example. Let's convert all fractions to improper ones and perform the required operations. In order not to bore the reader, I will omit some obvious calculations. We have:

Due to the fact that the numerator and denominator of the basic fractions contain sums, the rule for writing multi-story fractions is observed automatically. Also, in the last example, we intentionally left 46/1 in fraction form to perform division.

I will also note that in both examples the fraction bar actually replaces the parentheses: first of all, we found the sum, and only then the quotient.

Some will say that the transition to improper fractions in the second example was clearly redundant. Perhaps this is true. But by doing this we insure ourselves against mistakes, because next time the example may turn out to be much more complicated. Choose for yourself what is more important: speed or reliability.

This article examines operations on fractions. Rules for addition, subtraction, multiplication, division or exponentiation of fractions of the form A B will be formed and justified, where A and B can be numbers, numerical expressions or expressions with variables. In conclusion, examples of solutions with detailed descriptions will be considered.

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Rules for performing operations with general numerical fractions

Numerical fractions general view have a numerator and denominator that contain natural numbers or numerical expressions. If we consider fractions such as 3 5, 2, 8 4, 1 + 2 3 4 (5 - 2), 3 4 + 7 8 2, 3 - 0, 8, 1 2 2, π 1 - 2 3 + π, 2 0, 5 ln 3, then it is clear that the numerator and denominator can have not only numbers, but also expressions of various types.

Definition 1

There are rules by which operations with ordinary fractions are carried out. It is also suitable for general fractions:

• When subtracting fractions with like denominators, only the numerators are added, and the denominator remains the same, namely: a d ± c d = a ± c d, the values ​​a, c and d ≠ 0 are some numbers or numerical expressions.
• When adding or subtracting a fraction with different denominators, it is necessary to reduce it to a common denominator, and then add or subtract the resulting fractions with the same exponents. Literally it looks like this: a b ± c d = a · p ± c · r s, where the values ​​a, b ≠ 0, c, d ≠ 0, p ≠ 0, r ≠ 0, s ≠ 0 are real numbers, and b · p = d · r = s . When p = d and r = b, then a b ± c d = a · d ± c · d b · d.
• When multiplying fractions, the action is performed with numerators, after which with denominators, then we get a b · c d = a · c b · d, where a, b ≠ 0, c, d ≠ 0 act as real numbers.
• When dividing a fraction by a fraction, we multiply the first by the second inverse, that is, we swap the numerator and denominator: a b: c d = a b · d c.

Rationale for the rules

Definition 2

There are the following mathematical points that you should rely on when calculating:

• the slash means the division sign;
• division by a number is treated as multiplication by its reciprocal value;
• application of the property of operations with real numbers;
• application of the basic property of fractions and numerical inequalities.

With their help, you can perform transformations of the form:

a d ± c d = a · d - 1 ± c · d - 1 = a ± c · d - 1 = a ± c d ; a b ± c d = a · p b · p ± c · r d · r = a · p s ± c · e s = a · p ± c · r s ; a b · c d = a · d b · d · b · c b · d = a · d · a · d - 1 · b · c · b · d - 1 = = a · d · b · c · b · d - 1 · b · d - 1 = a · d · b · c b · d · b · d - 1 = = (a · c) · (b · d) - 1 = a · c b · d

Examples

In the previous paragraph it was said about operations with fractions. It is after this that the fraction needs to be simplified. This topic was discussed in detail in the paragraph on converting fractions.

First, let's look at an example of adding and subtracting fractions with the same denominator.

Example 1

Given the fractions 8 2, 7 and 1 2, 7, then according to the rule it is necessary to add the numerator and rewrite the denominator.

Solution

Then we get a fraction of the form 8 + 1 2, 7. After performing the addition, we obtain a fraction of the form 8 + 1 2, 7 = 9 2, 7 = 90 27 = 3 1 3. So, 8 2, 7 + 1 2, 7 = 8 + 1 2, 7 = 9 2, 7 = 90 27 = 3 1 3.

Answer: 8 2 , 7 + 1 2 , 7 = 3 1 3

There is another solution. To begin with, we switch to the form of an ordinary fraction, after which we perform a simplification. It looks like this:

8 2 , 7 + 1 2 , 7 = 80 27 + 10 27 = 90 27 = 3 1 3

Example 2

Let's subtract from 1 - 2 3 · log 2 3 · log 2 5 + 1 a fraction of the form 2 3 3 · log 2 3 · log 2 5 + 1 .

Since equal denominators are given, it means that we are calculating a fraction with the same denominator. We get that

1 - 2 3 log 2 3 log 2 5 + 1 - 2 3 3 log 2 3 log 2 5 + 1 = 1 - 2 - 2 3 3 log 2 3 log 2 5 + 1

There are examples of calculating fractions with different denominators. An important point is reduction to a common denominator. Without this we will not be able to carry out further actions with fractions.

The process is vaguely reminiscent of reduction to a common denominator. That is, the least common divisor in the denominator is searched for, after which the missing factors are added to the fractions.

If the fractions being added do not have common factors, then their product can become one.

Example 3

Let's look at the example of adding fractions 2 3 5 + 1 and 1 2.

Solution

In this case, the common denominator is the product of the denominators. Then we get that 2 · 3 5 + 1. Then, when setting additional factors, we have that for the first fraction it is equal to 2, and for the second it is 3 5 + 1. After multiplication, the fractions are reduced to the form 4 2 · 3 5 + 1. The general reduction of 1 2 will be 3 5 + 1 2 · 3 5 + 1. We add the resulting fractional expressions and get that

2 3 5 + 1 + 1 2 = 2 2 2 3 5 + 1 + 1 3 5 + 1 2 3 5 + 1 = = 4 2 3 5 + 1 + 3 5 + 1 2 3 5 + 1 = 4 + 3 5 + 1 2 3 5 + 1 = 5 + 3 5 2 3 5 + 1

Answer: 2 3 5 + 1 + 1 2 = 5 + 3 5 2 3 5 + 1

When we are dealing with general fractions, then we usually do not talk about the lowest common denominator. It is unprofitable to take the product of the numerators as the denominator. First you need to check if there is a number that is less in value than their product.

Example 4

Let's consider the example of 1 6 · 2 1 5 and 1 4 · 2 3 5, when their product is equal to 6 · 2 1 5 · 4 · 2 3 5 = 24 · 2 4 5. Then we take 12 · 2 3 5 as the common denominator.

Let's look at examples of multiplying general fractions.

Example 5

To do this, you need to multiply 2 + 1 6 and 2 · 5 3 · 2 + 1.

Solution

Following the rule, it is necessary to rewrite and write the product of the numerators as a denominator. We get that 2 + 1 6 2 5 3 2 + 1 2 + 1 2 5 6 3 2 + 1. Once a fraction has been multiplied, you can make reductions to simplify it. Then 5 · 3 3 2 + 1: 10 9 3 = 5 · 3 3 2 + 1 · 9 3 10.

Using the rule for transition from division to multiplication by a reciprocal fraction, we obtain a fraction that is the reciprocal of the given one. To do this, the numerator and denominator are swapped. Let's look at an example:

5 3 3 2 + 1: 10 9 3 = 5 3 3 2 + 1 9 3 10

Then they must multiply and simplify the resulting fraction. If necessary, get rid of irrationality in the denominator. We get that

5 3 3 2 + 1: 10 9 3 = 5 3 3 9 3 10 2 + 1 = 5 2 10 2 + 1 = 3 2 2 + 1 = 3 2 - 1 2 2 + 1 2 - 1 = 3 2 - 1 2 2 2 - 1 2 = 3 2 - 1 2

Answer: 5 3 3 2 + 1: 10 9 3 = 3 2 - 1 2

This paragraph is applicable when a number or numerical expression can be represented as a fraction with a denominator equal to 1, then the operation with such a fraction is considered a separate paragraph. For example, the expression 1 6 · 7 4 - 1 · 3 shows that the root of 3 can be replaced by another 3 1 expression. Then this entry will look like multiplying two fractions of the form 1 6 · 7 4 - 1 · 3 = 1 6 · 7 4 - 1 · 3 1.

Performing Operations on Fractions Containing Variables

The rules discussed in the first article are applicable to operations with fractions containing variables. Consider the subtraction rule when the denominators are the same.

It is necessary to prove that A, C and D (D not equal to zero) can be any expressions, and the equality A D ± C D = A ± C D is equivalent to its domain acceptable values.

It is necessary to take a set of ODZ variables. Then A, C, D must take the corresponding values ​​a 0 , c 0 and d 0. Substitution of the form A D ± C D results in a difference of the form a 0 d 0 ± c 0 d 0 , where, using the addition rule, we obtain a formula of the form a 0 ± c 0 d 0 . If we substitute the expression A ± C D, then we get the same fraction of the form a 0 ± c 0 d 0. From here we conclude that the selected value that satisfies the ODZ, A ± C D and A D ± C D are considered equal.

For any value of the variables, these expressions will be equal, that is, they are called identically equal. This means that this expression is considered a provable equality of the form A D ± C D = A ± C D .

Examples of adding and subtracting fractions with variables

When you have the same denominators, you only need to add or subtract the numerators. This fraction can be simplified. Sometimes you have to work with fractions that are identically equal, but at first glance this is not noticeable, since some transformations must be performed. For example, x 2 3 x 1 3 + 1 and x 1 3 + 1 2 or 1 2 sin 2 α and sin a cos a. Most often, a simplification of the original expression is required in order to see the same denominators.

Example 6

Calculate: 1) x 2 + 1 x + x - 2 - 5 - x x + x - 2, 2) l g 2 x + 4 x · (l g x + 2) + 4 · l g x x · (l g x + 2) , x - 1 x - 1 + x x + 1 .

Solution

1. To make the calculation, you need to subtract fractions that have the same denominator. Then we get that x 2 + 1 x + x - 2 - 5 - x x + x - 2 = x 2 + 1 - 5 - x x + x - 2 . After which you can expand the brackets and add similar terms. We get that x 2 + 1 - 5 - x x + x - 2 = x 2 + 1 - 5 + x x + x - 2 = x 2 + x - 4 x + x - 2
2. Since the denominators are the same, all that remains is to add the numerators, leaving the denominator: l g 2 x + 4 x (l g x + 2) + 4 l g x x (l g x + 2) = l g 2 x + 4 + 4 x (l g x + 2)
   The addition has been completed. It can be seen that it is possible to reduce the fraction. Its numerator can be folded using the formula for the square of the sum, then we get (l g x + 2) 2 from abbreviated multiplication formulas. Then we get that
   l g 2 x + 4 + 2 l g x x (l g x + 2) = (l g x + 2) 2 x (l g x + 2) = l g x + 2 x
3. Given fractions of the form x - 1 x - 1 + x x + 1 with different denominators. After the transformation, you can move on to addition.

Let's consider a twofold solution.

The first method is that the denominator of the first fraction is factorized using squares, with its subsequent reduction. We get a fraction of the form

x - 1 x - 1 = x - 1 (x - 1) x + 1 = 1 x + 1

So x - 1 x - 1 + x x + 1 = 1 x + 1 + x x + 1 = 1 + x x + 1 .

In this case, it is necessary to get rid of irrationality in the denominator.

1 + x x + 1 = 1 + x x - 1 x + 1 x - 1 = x - 1 + x x - x x - 1

The second method is to multiply the numerator and denominator of the second fraction by the expression x - 1. Thus, we get rid of irrationality and move on to adding fractions with the same denominator. Then

x - 1 x - 1 + x x + 1 = x - 1 x - 1 + x x - 1 x + 1 x - 1 = = x - 1 x - 1 + x x - x x - 1 = x - 1 + x · x - x x - 1

Answer: 1) x 2 + 1 x + x - 2 - 5 - x x + x - 2 = x 2 + x - 4 x + x - 2, 2) l g 2 x + 4 x · (l g x + 2) + 4 · l g x x · (l g x + 2) = l g x + 2 x, 3) x - 1 x - 1 + x x + 1 = x - 1 + x · x - x x - 1 .

In the last example we found that reduction to a common denominator is inevitable. To do this, you need to simplify the fractions. When adding or subtracting, you always need to look for a common denominator, which looks like the product of the denominators with additional factors added to the numerators.

Example 7

Calculate the values ​​of the fractions: 1) x 3 + 1 x 7 + 2 2, 2) x + 1 x ln 2 (x + 1) (2 x - 4) - sin x x 5 ln (x + 1) (2 x - 4) , 3) ​​1 cos 2 x - x + 1 cos 2 x + 2 cos x x + x

Solution

1. The denominator does not require any complex calculations, so you need to choose their product of the form 3 x 7 + 2 · 2, then choose x 7 + 2 · 2 for the first fraction as an additional factor, and 3 for the second. When multiplying, we get a fraction of the form x 3 + 1 x 7 + 2 2 = x x 7 + 2 2 3 x 7 + 2 2 + 3 1 3 x 7 + 2 2 = = x x 7 + 2 2 + 3 3 x 7 + 2 2 = x x 7 + 2 2 x + 3 3 x 7 + 2 2
2. It can be seen that the denominators are presented in the form of a product, which means that additional transformations are unnecessary. The common denominator will be considered to be a product of the form x 5 · ln 2 x + 1 · 2 x - 4 . Hence x 4 is an additional factor to the first fraction, and ln(x + 1) to the second. Then we subtract and get:
   x + 1 x · ln 2 (x + 1) · 2 x - 4 - sin x x 5 · ln (x + 1) · 2 x - 4 = = x + 1 · x 4 x 5 · ln 2 (x + 1 ) · 2 x - 4 - sin x · ln x + 1 x 5 · ln 2 (x + 1) · (2 ​​x - 4) = = x + 1 · x 4 - sin x · ln (x + 1) x 5 · ln 2 (x + 1) · (2 ​​x - 4) = x · x 4 + x 4 - sin x · ln (x + 1) x 5 · ln 2 (x + 1) · (2 ​​x - 4 )
3. This example makes sense when working with fraction denominators. It is necessary to apply the formulas for the difference of squares and the square of the sum, since they will make it possible to move on to an expression of the form 1 cos x - x · cos x + x + 1 (cos x + x) 2. It can be seen that the fractions are reduced to a common denominator. We get that cos x - x · cos x + x 2 .

Then we get that

1 cos 2 x - x + 1 cos 2 x + 2 cos x x + x = = 1 cos x - x cos x + x + 1 cos x + x 2 = = cos x + x cos x - x cos x + x 2 + cos x - x cos x - x cos x + x 2 = = cos x + x + cos x - x cos x - x cos x + x 2 = 2 cos x cos x - x cos x + x 2

Answer:

1) x 3 + 1 x 7 + 2 2 = x x 7 + 2 2 x + 3 3 x 7 + 2 2, 2) x + 1 x ln 2 (x + 1) 2 x - 4 - sin x x 5 · ln (x + 1) · 2 x - 4 = = x · x 4 + x 4 - sin x · ln (x + 1) x 5 · ln 2 (x + 1) · ( 2 x - 4) , 3) ​​1 cos 2 x - x + 1 cos 2 x + 2 · cos x · x + x = 2 · cos x cos x - x · cos x + x 2 .

Examples of multiplying fractions with variables

When multiplying fractions, the numerator is multiplied by the numerator and the denominator by the denominator. Then you can apply the reduction property.

Example 8

Multiply the fractions x + 2 · x x 2 · ln x 2 · ln x + 1 and 3 · x 2 1 3 · x + 1 - 2 sin 2 · x - x.

Solution

Multiplication needs to be done. We get that

x + 2 x x 2 ln x 2 ln x + 1 3 x 2 1 3 x + 1 - 2 sin (2 x - x) = = x - 2 x 3 x 2 1 3 x + 1 - 2 x 2 ln x 2 ln x + 1 sin (2 x - x)

The number 3 is moved to the first place for the convenience of calculations, and you can reduce the fraction by x 2, then we get an expression of the form

3 x - 2 x x 1 3 x + 1 - 2 ln x 2 ln x + 1 sin (2 x - x)

Answer: x + 2 x x 2 ln x 2 ln x + 1 3 x 2 1 3 x + 1 - 2 sin (2 x - x) = 3 x - 2 x x 1 3 x + 1 - 2 ln x 2 · ln x + 1 · sin (2 · x - x) .

Division

Division of fractions is similar to multiplication, since the first fraction is multiplied by the second reciprocal. If we take for example the fraction x + 2 x x 2 ln x 2 ln x + 1 and divide by 3 x 2 1 3 x + 1 - 2 sin 2 x - x, then it can be written as

x + 2 · x x 2 · ln x 2 · ln x + 1: 3 · x 2 1 3 · x + 1 - 2 sin (2 · x - x) , then replace with a product of the form x + 2 · x x 2 · ln x 2 ln x + 1 3 x 2 1 3 x + 1 - 2 sin (2 x - x)

Exponentiation

Let's move on to considering operations with general fractions with exponentiation. If you have a degree with natural indicator, then the action is considered as multiplication of equal fractions. But it is recommended to use general approach, based on the properties of degrees. Any expressions A and C, where C is not identically equal to zero, and any real r on the ODZ for an expression of the form A C r the equality A C r = A r C r is valid. The result is a fraction raised to a power. For example, consider:

x 0, 7 - π · ln 3 x - 2 - 5 x + 1 2, 5 = = x 0, 7 - π · ln 3 x - 2 - 5 2, 5 x + 1 2, 5

Procedure for performing operations with fractions

Operations on fractions are performed according to certain rules. In practice, we notice that an expression may contain several fractions or fractional expressions. Then it is necessary to perform all actions in strict order: raise to a power, multiply, divide, then add and subtract. If there are parentheses, the first action is performed in them.

Example 9

Calculate 1 - x cos x - 1 c o s x · 1 + 1 x .

Solution

Since we have the same denominator, then 1 - x cos x and 1 c o s x, but subtractions cannot be performed according to the rule; first, the actions in parentheses are performed, then multiplication, and then addition. Then when calculating we get that

1 + 1 x = 1 1 + 1 x = x x + 1 x = x + 1 x

When substituting the expression into the original one, we get that 1 - x cos x - 1 cos x · x + 1 x. When multiplying fractions we have: 1 cos x · x + 1 x = x + 1 cos x · x. Having made all the substitutions, we get 1 - x cos x - x + 1 cos x · x. Now you need to work with fractions that have different denominators. We get:

x · 1 - x cos x · x - x + 1 cos x · x = x · 1 - x - 1 + x cos x · x = = x - x - x - 1 cos x · x = - x + 1 cos x x

Answer: 1 - x cos x - 1 c o s x · 1 + 1 x = - x + 1 cos x · x .

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This article is a general look at operating with fractions. Here we will formulate and justify the rules for addition, subtraction, multiplication, division and exponentiation of fractions of the general form A/B, where A and B are some numbers, numerical expressions or expressions with variables. As usual, we will provide the material with explanatory examples with detailed descriptions of solutions.

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Rules for performing operations with general numerical fractions

Let's agree on numerical fractions general form to understand fractions in which the numerator and/or denominator can be represented not only natural numbers, but also other numbers or numerical expressions. For clarity, here are a few examples of such fractions: , .

We know the rules by which they are carried out. Using the same rules, you can perform operations with general fractions:

Rationale for the rules

To justify the validity of the rules for performing operations with numerical fractions of a general form, you can start from the following points:

• The slash is essentially a division sign,
• division by some nonzero number can be considered as multiplication by the inverse of the divisor (this immediately explains the rule dividing fractions),
• properties of operations with real numbers,
• and its general understanding,

They allow you to carry out the following transformations that justify the rules of addition, subtraction of fractions with like and unlike denominators, as well as the rule of multiplication of fractions:

Examples

Let us give examples of performing operations with general fractions according to the rules learned in the previous paragraph. Let's say right away that usually after performing operations with fractions, the resulting fraction requires simplification, and the process of simplifying a fraction is often more complicated than performing previous actions. We will not dwell in detail on simplifying fractions (the corresponding transformations are discussed in the article transforming fractions), so as not to be distracted from the topic that interests us.

Let's start with examples of adding and subtracting fractions with like denominators. First, let's add the fractions and . Obviously the denominators are equal. According to the corresponding rule, we write down a fraction whose numerator is equal to the sum of the numerators of the original fractions, and leave the denominator the same, we have. The addition is done, all that remains is to simplify the resulting fraction: . So, .

The solution could have been handled differently: first make the transition to ordinary fractions, and then carry out the addition. With this approach we have .

Now let's subtract from the fraction fraction . The denominators of the fractions are equal, therefore, we follow the rule for subtracting fractions with the same denominators:

Let's move on to examples of adding and subtracting fractions with different denominators. The main difficulty here is bringing fractions to a common denominator. For general fractions, this is a rather extensive topic; we will examine it in detail in a separate article. bringing fractions to a common denominator. Now let's limit ourselves to a couple general recommendations, since in this moment we are more interested in the technique of performing operations with fractions.

In general, the process is similar to reducing ordinary fractions to a common denominator. That is, the denominators are presented in the form of products, then all the factors from the denominator of the first fraction are taken and the missing factors from the denominator of the second fraction are added to them.

When the denominators of fractions being added or subtracted do not have common factors, then it is logical to take their product as the common denominator. Let's give an example.

Let's say we need to perform addition of fractions and 1/2. Here, as a common denominator, it is logical to take the product of the denominators of the original fractions, that is, . In this case, the additional factor for the first fraction will be 2. After multiplying the numerator and denominator by it, the fraction will take the form . And for the second fraction, the additional factor is the expression. With its help, the fraction 1/2 is reduced to the form . All that remains is to add the resulting fractions with the same denominators. Here's a summary of the entire solution:

In the case of general fractions, we are no longer talking about the lowest common denominator, to which ordinary fractions are usually reduced. Although in this matter it is still advisable to strive for some minimalism. By this we want to say that you should not immediately take the product of the denominators of the original fractions as a common denominator. For example, it is not at all necessary to take the common denominator of fractions and the product . Here we can take .

Let's move on to examples of multiplying general fractions. Let's multiply fractions and . The rule for performing this action instructs us to write down a fraction, the numerator of which is the product of the numerators of the original fractions, and the denominator is the product of the denominators. We have . Here, as in many other cases when multiplying fractions, you can reduce the fraction: .

The rule for dividing fractions allows you to move from division to multiplication by the reciprocal fraction. Here you need to remember that in order to get the inverse of a given fraction, you need to swap the numerator and denominator of the given fraction. Here is an example of the transition from division of general numerical fractions to multiplication: . All that remains is to perform the multiplication and simplify the resulting fraction (if necessary, see the transformation of irrational expressions):

Concluding the information in this paragraph, recall that any number or numerical expression can be represented as a fraction with a denominator 1, therefore, addition, subtraction, multiplication and division of numbers and fractions can be considered as performing the corresponding operation with fractions, one of which has one in the denominator . For example, replacing in the expression root of three by a fraction, we move from multiplying a fraction by a number to multiplying two fractions: .

Doing things with fractions that contain variables

The rules from the first part of this article also apply to performing operations with fractions that contain variables. Let's justify the first of them - the rule for adding and subtracting fractions with identical denominators, the rest are proven in absolutely the same way.

Let us prove that for any expressions A, C and D (D is not identically equal to zero) the equality holds on its range of permissible values ​​of variables.

Let's take a certain set of variables from the ODZ. Let the expressions A, C and D take the values ​​a 0, c 0 and d 0 for these values ​​of the variables. Then substituting the values ​​of variables from the selected set into the expression turns it into a sum (difference) of numerical fractions with like denominators of the form , which, according to the rule of addition (subtraction) of numerical fractions with like denominators, is equal to . But substituting the values ​​of variables from the selected set into the expression turns it into the same fraction. This means that for the selected set of variable values ​​from the ODZ, the values ​​of the expressions and are equal. It is clear that the values ​​of the indicated expressions will be equal for any other set of values ​​of variables from the ODZ, which means that the expressions and are identically equal, that is, the equality being proved is true .

Examples of adding and subtracting fractions with variables

When the denominators of the fractions being added or subtracted are the same, then everything is quite simple - the numerators are added or subtracted, but the denominator remains the same. It is clear that the fraction obtained after this is simplified if necessary and possible.

Note that sometimes the denominators of fractions differ only at first glance, but in fact they are identically equal expressions, for example, and , or and . And sometimes it is enough to simplify the original fractions so that their identical denominators “appear.”

Example.

, b) , V) .

Solution.

a) We need to subtract fractions with like denominators. According to the corresponding rule, we leave the denominator the same and subtract the numerators, we have . The action has been completed. But you can also open the parentheses in the numerator and present similar terms: .

b) Obviously, the denominators of the fractions being added are the same. Therefore, we add up the numerators and leave the denominator the same: . Addition completed. But it is easy to see that the resulting fraction can be reduced. Indeed, the numerator of the resulting fraction can be collapsed using the formula square of the sum as (lgx+2) 2 (see formulas for abbreviated multiplication), thus the following transformations take place: .

c) Fractions in sum have different denominators. But, having transformed one of the fractions, you can move on to adding fractions with the same denominators. We will show two solutions.

First way. The denominator of the first fraction can be factorized using the difference of squares formula, and then reduce this fraction: . Thus, . It still doesn’t hurt to free yourself from irrationality in the denominator of the fraction: .

Second way. Multiplying the numerator and denominator of the second fraction by (this expression does not go to zero for any value of the variable x from the ODZ for the original expression) allows you to achieve two goals at once: free yourself from irrationality and move on to adding fractions with the same denominators. We have

Answer:

A) , b) , V) .

The last example brought us to the question of reducing fractions to a common denominator. There we almost accidentally arrived at the same denominators by simplifying one of the added fractions. But in most cases, when adding and subtracting fractions with different denominators, you have to purposefully bring the fractions to a common denominator. To do this, usually the denominators of fractions are presented in the form of products, all the factors from the denominator of the first fraction are taken and the missing factors from the denominator of the second fraction are added to them.

Example.

Perform operations with fractions: a) , b) , c) .

Solution.

a) There is no need to do anything with the denominators of the fractions. As a common denominator we take the product . In this case, the additional factor for the first fraction is the expression, and for the second fraction - the number 3. These additional factors bring the fractions to a common denominator, which later allows us to perform the action we need, we have

b) In this example, the denominators are already represented as products and do not require any additional transformations. Obviously, the factors in the denominators differ only in exponents, therefore, as a common denominator we take the product of the factors with the highest exponents, that is, . Then the additional factor for the first fraction will be x 4, and for the second – ln(x+1) . Now we're ready to subtract fractions:

c) And in this case, first we will work with the denominators of fractions. The formulas for the difference of squares and the square of the sum allow you to move from the original sum to the expression . Now it is clear that these fractions can be reduced to a common denominator . With this approach, the solution will have next view:

Answer:

A)

b)

V)

Examples of multiplying fractions with variables

Multiplying fractions produces a fraction whose numerator is the product of the numerators of the original fractions, and the denominator is the product of the denominators. Here, as you can see, everything is familiar and simple, and we can only add that the fraction obtained as a result of this action often turns out to be reducible. In these cases, it is reduced, unless, of course, it is necessary and justified.

Multiplying and dividing fractions.

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

This operation is much nicer than addition-subtraction! Because it's easier. As a reminder, to multiply a fraction by a fraction, you need to multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). That is:

For example:

Everything is extremely simple. And please don't look for a common denominator! There is no need for him here...

To divide a fraction by a fraction, you need to reverse second(this is important!) fraction and multiply them, i.e.:

For example:

If you come across multiplication or division with integers and fractions, it’s okay. As with addition, we make a fraction from a whole number with one in the denominator - and go ahead! For example:

In high school, you often have to deal with three-story (or even four-story!) fractions. For example:

How can I make this fraction look decent? Yes, very simple! Use two-point division:

But don't forget about the order of division! Unlike multiplication, this is very important here! Of course, we will not confuse 4:2 or 2:4. But it’s easy to make a mistake in a three-story fraction. Please note for example:

In the first case (expression on the left):

In the second (expression on the right):

Do you feel the difference? 4 and 1/9!

What determines the order of division? Either with brackets, or (as here) with the length of horizontal lines. Develop your eye. And if there are no brackets or dashes, like:

then divide and multiply in order, from left to right!

And another very simple and important technique. In actions with degrees, it will be so useful to you! Let's divide one by any fraction, for example, by 13/15:

The shot has turned over! And this always happens. When dividing 1 by any fraction, the result is the same fraction, only upside down.

That's it for operations with fractions. The thing is quite simple, but it gives more than enough errors. Note practical advice, and there will be fewer of them (errors)!

Practical tips:

1. The most important thing when working with fractional expressions is accuracy and attentiveness! Is not common words, not good wishes! This is a dire necessity! Do all calculations on the Unified State Exam as a full-fledged task, focused and clear. It’s better to write two extra lines in your draft than to mess up when doing mental calculations.

2. In examples with different types fractions - go to ordinary fractions.

3. We reduce all fractions until they stop.

4. We reduce multi-level fractional expressions to ordinary ones using division through two points (we follow the order of division!).

5. Divide a unit by a fraction in your head, simply turning the fraction over.

Here are the tasks that you must definitely complete. Answers are given after all tasks. Use the materials on this topic and practical tips. Estimate how many examples you were able to solve correctly. The first time! Without a calculator! And draw the right conclusions...

Remember - the correct answer is received from the second (especially the third) time does not count! Such is the harsh life.

So, solve in exam mode ! This is already preparation for the Unified State Exam, by the way. We solve the example, check it, solve the next one. We decided everything - checked again from first to last. But only Then look at the answers.

Calculate:

Have you decided?

We are looking for answers that match yours. I deliberately wrote them down in disarray, away from temptation, so to speak... Here they are, the answers, written with semicolons.

0; 17/22; 3/4; 2/5; 1; 25.

Now we draw conclusions. If everything worked out, I’m happy for you! Basic calculations with fractions are not your problem! You can do more serious things. If not...

So you have one of two problems. Or both at once.) Lack of knowledge and (or) inattention. But this solvable Problems.

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!)

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