Fractional expressions are difficult for a child to understand. Most people have difficulties with. When studying the topic “adding fractions with whole numbers,” the child falls into a stupor, finding it difficult to solve the problem. In many examples, before performing an action, a series of calculations must be performed. For example, convert fractions or convert an improper fraction to a proper fraction.
Let’s explain it clearly to the child. Let's take three apples, two of which will be whole, and cut the third into 4 parts. Separate one slice from the cut apple, and place the remaining three next to two whole fruits. We get ¼ of an apple on one side and 2 ¾ on the other. If we combine them, we get three apples. Let's try to reduce 2 ¾ apples by ¼, that is, remove another slice, we get 2 2/4 apples.
Let's take a closer look at operations with fractions that contain integers:
First, let's remember the calculation rule for fractional expressions with a common denominator:
At first glance, everything is easy and simple. But this only applies to expressions that do not require conversion.
How to find the value of an expression where the denominators are different
In some tasks you need to find the meaning of an expression where the denominators are different. Let's look at a specific case:
3 2/7+6 1/3
Let's find the value given expression, for this we find for two fractions common denominator.
For the numbers 7 and 3, this is 21. We leave the integer parts the same, and bring the fractional parts to 21, for this we multiply the first fraction by 3, the second by 7, we get:
6/21+7/21, do not forget that whole parts cannot be converted. As a result, we get two fractions with the same denominator and calculate their sum:
3 6/21+6 7/21=9 15/21
What if the result of addition is an improper fraction that already has an integer part:
2 1/3+3 2/3
IN in this case We add up the whole parts and fractional parts, we get:
5 3/3, as you know, 3/3 is one, which means 2 1/3+3 2/3=5 3/3=5+1=6
Finding the sum is all clear, let’s look at the subtraction:
From all that has been said, the rule of action over mixed numbers, which sounds like this:
- If from fractional expression it is necessary to subtract an integer, there is no need to represent the second number as a fraction, it is enough to perform the operation only on the integer parts.
Let's try to calculate the meaning of the expressions ourselves:
Let's sort it out more example under the letter "m":
4 5/11-2 8/11, the numerator of the first fraction is less than the second. To do this, we borrow one integer from the first fraction, we get,
3 5/11+11/11=3 whole 16/11, subtract the second from the first fraction:
3 16/11-2 8/11=1 whole 8/11
- Be careful when completing the task, do not forget to convert improper fractions into mixed fractions, highlighting the whole part. To do this, you need to divide the value of the numerator by the value of the denominator, what you get takes the place of the whole part, the remainder will be the numerator, for example:
19/4=4 ¾, let’s check: 4*4+3=19, the denominator 4 remains unchanged.
Let's summarize:
Before you begin to complete a task related to fractions, you need to analyze what kind of expression it is, what transformations need to be made on the fraction in order for the solution to be correct. Look for more rational way solutions. Don't go the hard way. Plan all the actions, solve them first in draft form, then transfer them to your school notebook.
To avoid confusion when solving fractional expressions, you must follow the rule of consistency. Decide everything carefully, without rushing.
Oh those fractions! IN high school In math lessons, it is arithmetic operations with fractions and problems where numbers with numerators and denominators flash in the conditions that become an obstacle that many schoolchildren have difficulty overcoming. Memorization and use is enough simple rules, which govern operations with fractions, for some students the obstacle to good grades in mathematics. So how do you solve problems with fractions? This is possible if you understand correctly what a fraction is.
For clear example Let's take a regular cake. You are expecting seven guests for the holiday. You have only one cake. This means that it must be divided into eight (guests plus the birthday person). You cut the cake into equal parts. Each of these parts is only 1/8 of the whole pie. The result is a simple natural fraction, where 1 is the numerator and 8 is the denominator. One of the guests refused the pie, and you decided to take another piece for yourself. Now there are 2 pieces of eight parts of the pie, or 2/8.
What if all your guests are on diets, losing weight and don’t want to eat cake? Then you get eight parts out of eight (8/8), that is, one whole cake!
Fractions where the numerator is less than the denominator, are called correct. And those with a larger numerator are incorrect.
Problems with natural fractions
Tasks that involve natural fractions, most often involve actions with them. Most easy option Such a problem is finding the fraction of a number that is expressed as a fraction. You were given 6 kilograms of apples. You should leave 2/3 of them for preparing the pie filling. We multiply 6 by 2, then divide by 3. As a result, we have 4 kilos needed for the filling.
If it's worth difficult task find a number by its part, multiply the part of the number by the fraction, swapping the numerator and denominator. Here there are 6 kilograms of apples. This is 3/5 of total number apples collected from your apple tree. This means that we quickly multiply 6 by 5 and divide by 3. It comes out to 10 kilograms.
How are fractions divided and multiplied? The rules here are simple. When we multiply a fraction by a fraction, we perform operations with numerators and denominators. Let's say you need to multiply 2/3 by 5/6. We multiply the number 2 by 5, and multiply 3 by 6. Result: 10/18. If you need to multiply a fraction by a whole number, simply multiply the number itself and the numerator of the fraction. So 3*4/7=12/7. Convert the fraction to the correct one: 12/7=1 and 5/7.
We can easily replace division of fractions with multiplication. Need to divide 5/6 by 2/3? This means that we leave the first fraction 5/6 unchanged, and in the second we swap the numerator and denominator. 5/6:2/3=5/6*3/2=15/12. Similar rules exist for dividing a natural number by a fraction. 2:4/7= 2*7/4=14/4. If we divide a fraction by natural number, then we multiply the denominator and the number itself. 4/7:2=4/14.
It is more difficult to perform subtraction and addition with fractions where the denominators are different. If you need to add the fraction 2/8 to 3/8, this is easier. Add the numerators, leaving the denominators unchanged. It comes out 5/8. With subtraction, everything is the same, where the smaller one is subtracted from the larger numerator.
How to solve problems with fractions where the denominators are different? Of course, first bring them to one. For example, you need to add 5/8 and 2/3. Using the selection method, we are looking for a number that is divisible by both 8 and 3. This number is 24. To make a fraction from 5/8 with a denominator of 24, divide 24 by 8. The number we get is 3. Multiply the numerator by 3. As a result, 5/8 equals 15/24. We do the same with 2/3, getting 16/24. You can then add and subtract the denominators.
We received an improper fraction 31/24. 24/24 is one whole number. Subtract the denominator from the numerator. It turns out 1 whole and 7/24.
What to do when you need to subtract a part from a whole number? You have three cakes that you need to cut into five pieces each and give 2/5 to someone you know. 3 is 15 divided by five. So you have 15/5 cake. Subtract 2 from 15, it turns out that you were left with 13/5 of the cake, or 2 whole and 3/5.
This is how you can solve problems with fractions. The main thing is to remember that you cannot subtract a larger numerator from a smaller one!
The numerator, and that which is divided by is the denominator.
To write a fraction, first write the numerator, then draw a horizontal line under the number, and write the denominator below the line. The horizontal line separating the numerator and denominator is called a fraction line. Sometimes it is depicted as an oblique "/" or "∕". In this case, the numerator is written to the left of the line, and the denominator to the right. So, for example, the fraction “two thirds” will be written as 2/3. For clarity, the numerator is usually written at the top of the line, and the denominator at the bottom, that is, instead of 2/3 you can find: ⅔.
To calculate the product of fractions, first multiply the numerator of one fractions to the numerator is different. Write the result in the numerator of the new fractions. After this, multiply the denominators. Enter the total value in the new fractions. For example, 1/3? 1/5 = 1/15 (1 × 1 = 1; 3 × 5 = 15).
To divide one fraction by another, first multiply the numerator of the first by the denominator of the second. Do the same with the second fraction (divisor). Or, before performing all the actions, first “flip” the divisor, if it is more convenient for you: the denominator should be in place of the numerator. Then multiply the denominator of the dividend by the new denominator of the divisor and multiply the numerators. For example, 1/3: 1/5 = 5/3 = 1 2/3 (1 ? 5 = 5; 3 ? 1 = 3).
Sources:
- Basic fraction problems
Fractional numbers can be expressed in in different forms exact value of the quantity. You can do the same math operations with fractions as you can with whole numbers: subtraction, addition, multiplication, and division. To learn to decide fractions, we must remember some of their features. They depend on the type fractions, the presence of an integer part, a common denominator. Some arithmetic operations require the fractional part of the result to be reduced after execution.
You will need
- - calculator
Instructions
Take a close look at the numbers. If among the fractions there are decimals and irregular ones, sometimes it is more convenient to first perform operations with decimals, and then convert them to the irregular form. Can you translate fractions in this form initially, writing the value after the decimal point in the numerator and putting 10 in the denominator. If necessary, reduce the fraction by dividing the numbers above and below by one divisor. Fractions in which an integer part is isolated must be converted to the wrong form by multiplying it by the denominator and adding the numerator to the result. Given value will become the new numerator fractions. To select a whole part from an initially incorrect one fractions, you need to divide the numerator by the denominator. Whole result write down from fractions. And the remainder of the division will become the new numerator, denominator fractions it does not change. For fractions with whole part it is possible to perform actions separately first for the integer and then for the fractional parts. For example, the sum of 1 2/3 and 2 ¾ can be calculated:
- Converting fractions to the wrong form:
- 1 2/3 + 2 ¾ = 5/3 + 11/4 = 20/12 + 33/12 = 53/12 = 4 5/12;
- Summing separately integers and fractional parts terms:
- 1 2/3 + 2 ¾ = (1+2) + (2/3 + ¾) = 3 +(8/12 + 9/12) = 3 + 12/17 = 3 + 1 5/12 = 4 5 /12.
Rewrite them using the “:” separator and continue regular division.
To receive final result Reduce the resulting fraction by dividing the numerator and denominator by one whole number, the largest possible in this case. In this case, there must be integers above and below the line.
Please note
Do not perform arithmetic with fractions whose denominators are different. Choose a number such that when you multiply the numerator and denominator of each fraction by it, the result is that the denominators of both fractions are equal.
When recording fractional numbers The dividend is written above the line. This quantity is designated as the numerator of the fraction. The divisor, or denominator, of the fraction is written below the line. For example, one and a half kilograms of rice as a fraction will be written as follows: 1 ½ kg of rice. If the denominator of a fraction is 10, the fraction is called a decimal. In this case, the numerator (dividend) is written to the right of the whole part, separated by a comma: 1.5 kg of rice. For ease of calculation, such a fraction can always be written in in the wrong form: 1 2/10 kg potatoes. To simplify, you can reduce the numerator and denominator values by dividing them by one integer. IN in this example may be divided by 2. The result will be 1 1/5 kg of potatoes. Make sure that the numbers you are going to perform arithmetic with are presented in the same form.
When a student moves to high school, mathematics is divided into 2 subjects: algebra and geometry. There are more and more concepts, the tasks are more and more difficult. Some people have difficulty understanding fractions. Missed the first lesson on this topic, and voila. fractions? A question that will torment throughout my school life.
The concept of an algebraic fraction
Let's start with a definition. Under algebraic fraction refers to the expressions P/Q, where P is the numerator and Q is the denominator. A number, a numerical expression, or a numerical-alphabetic expression can be hidden under a letter entry.
Before you wonder how to decide algebraic fractions, first you need to understand that similar expression- part of the whole.
As a rule, an integer is 1. The number in the denominator shows how many parts the unit is divided into. The numerator is needed to find out how many elements are taken. The fraction bar corresponds to the division sign. It is allowed to write a fractional expression as a mathematical operation “Division”. In this case, the numerator is the dividend, the denominator is the divisor.
Basic rule of common fractions
When students pass this topic at school, they are given examples to reinforce. To solve and find them correctly different ways from difficult situations, you need to apply the basic property of fractions.
It goes like this: If you multiply both the numerator and the denominator by the same number or expression (other than zero), the value of the common fraction does not change. A special case from of this rule is the division of both sides of an expression by the same number or polynomial. Such transformations are called identical equalities.
Below we will look at how to solve addition and subtraction of algebraic fractions, multiplying, dividing and reducing fractions.
Mathematical operations with fractions
Let's look at how to solve, the main property of an algebraic fraction, and how to apply it in practice. If you need to multiply two fractions, add them, divide one by another, or subtract, you must always follow the rules.
Thus, for the operation of addition and subtraction, an additional factor must be found in order to bring the expressions to a common denominator. If the fractions are initially given with the same expressions Q, then this paragraph should be omitted. Once the common denominator is found, how do you solve algebraic fractions? You need to add or subtract numerators. But! It must be remembered that if there is a “-” sign in front of the fraction, all signs in the numerator are reversed. Sometimes you should not make any substitutions and mathematical operations. It is enough to change the sign in front of the fraction.
The concept is often used as reducing fractions. This means the following: if the numerator and denominator are divided by an expression different from one (the same for both parts), then a new fraction is obtained. The dividend and divisor are smaller than before, but due to the basic rule of fractions they remain equal to the original example.
The purpose of this operation is to obtain a new irreducible expression. Decide this task it is possible if you reduce the numerator and denominator by the largest common divisor. The operation algorithm consists of two points:
- Finding gcd for both sides of the fraction.
- Dividing the numerator and denominator by the found expression and obtaining irreducible fraction, equal to the previous one.
Below is a table showing the formulas. For convenience, you can print it out and carry it with you in a notebook. However, so that in the future, when solving a test or exam, there will be no difficulties in the question of how to solve algebraic fractions, these formulas must be learned by heart.
Several examples with solutions
WITH theoretical point From a perspective, the question of how to solve algebraic fractions is considered. The examples given in the article will help you better understand the material.
1. Convert fractions and bring them to a common denominator.
2. Convert fractions and bring them to a common denominator.
After studying the theoretical part and considering practical issues there shouldn't be any more.
This article discusses how to find the values of mathematical expressions. Let's start with simple numerical expressions and then consider cases as their complexity increases. At the end we give an expression containing letter designations, brackets, roots, special mathematical signs, degrees, functions, etc. As per tradition, we will provide the entire theory with abundant and detailed examples.
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How to find the value of a numeric expression?
Numerical expressions, among other things, help describe the problem condition mathematical language. At all mathematical expressions can be either very simple, consisting of a pair of numbers and arithmetic symbols, or very complex, containing functions, powers, roots, parentheses, etc. As part of a task, it is often necessary to find the meaning of a particular expression. How to do this will be discussed below.
The simplest cases
These are cases where the expression contains nothing but numbers and arithmetic operations. To successfully find the values of such expressions, you will need knowledge of the order of performing arithmetic operations without parentheses, as well as the ability to perform operations with various numbers.
If the expression contains only numbers and arithmetic signs " + " , " · " , " - " , " ÷ " , then the actions are performed from left to right in the following order: first multiplication and division, then addition and subtraction. Let's give examples.
Example 1: The value of a numeric expression
Let you need to find the values of the expression 14 - 2 · 15 ÷ 6 - 3.
Let's do the multiplication and division first. We get:
14 - 2 15 ÷ 6 - 3 = 14 - 30 ÷ 6 - 3 = 14 - 5 - 3.
Now we carry out the subtraction and get the final result:
14 - 5 - 3 = 9 - 3 = 6 .
Example 2: The value of a numeric expression
Let's calculate: 0, 5 - 2 · - 7 + 2 3 ÷ 2 3 4 · 11 12.
First we perform fraction conversion, division and multiplication:
0, 5 - 2 · - 7 + 2 3 ÷ 2 3 4 · 11 12 = 1 2 - (- 14) + 2 3 ÷ 11 4 · 11 12
1 2 - (- 14) + 2 3 ÷ 11 4 11 12 = 1 2 - (- 14) + 2 3 4 11 11 12 = 1 2 - (- 14) + 2 9.
Now let's do some addition and subtraction. Let's group the fractions and bring them to a common denominator:
1 2 - (- 14) + 2 9 = 1 2 + 14 + 2 9 = 14 + 13 18 = 14 13 18 .
The required value has been found.
Expressions with parentheses
If an expression contains parentheses, they define the order of operations in that expression. The actions in brackets are performed first, and then all the others. Let's show this with an example.
Example 3: The value of a numeric expression
Let's find the value of the expression 0.5 · (0.76 - 0.06).
The expression contains parentheses, so we first perform the subtraction operation in parentheses, and only then the multiplication.
0.5 · (0.76 - 0.06) = 0.5 · 0.7 = 0.35.
The meaning of expressions containing parentheses within parentheses is found according to the same principle.
Example 4: The value of a numeric expression
Let's calculate the value 1 + 2 1 + 2 1 + 2 1 - 1 4.
We will perform actions starting from the innermost brackets, moving to the outer ones.
1 + 2 1 + 2 1 + 2 1 - 1 4 = 1 + 2 1 + 2 1 + 2 3 4
1 + 2 1 + 2 1 + 2 3 4 = 1 + 2 1 + 2 2, 5 = 1 + 2 6 = 13.
In finding the meanings of expressions with brackets, the main thing is to follow the sequence of actions.
Expressions with roots
Mathematical expressions whose values we need to find may contain root signs. Moreover, the expression itself may be under the root sign. What to do in this case? First you need to find the value of the expression under the root, and then extract the root from the number obtained as a result. If possible, it is better to get rid of roots in numerical expressions, replacing from with numeric values.
Example 5: The value of a numeric expression
Let's calculate the value of the expression with roots - 2 · 3 - 1 + 60 ÷ 4 3 + 3 · 2, 2 + 0, 1 · 0, 5.
First, we calculate the radical expressions.
2 3 - 1 + 60 ÷ 4 3 = - 6 - 1 + 15 3 = 8 3 = 2
2, 2 + 0, 1 0, 5 = 2, 2 + 0, 05 = 2, 25 = 1, 5.
Now you can calculate the value of the entire expression.
2 3 - 1 + 60 ÷ 4 3 + 3 2, 2 + 0, 1 0, 5 = 2 + 3 1, 5 = 6, 5
Often, finding the meaning of an expression with roots often requires first converting the original expression. Let's explain this with one more example.
Example 6: The value of a numeric expression
What is 3 + 1 3 - 1 - 1
As you can see, we do not have the opportunity to replace the root with an exact value, which complicates the counting process. However, in this case, you can apply the abbreviated multiplication formula.
3 + 1 3 - 1 = 3 - 1 .
Thus:
3 + 1 3 - 1 - 1 = 3 - 1 - 1 = 1 .
Expressions with powers
If an expression contains powers, their values must be calculated before proceeding with all other actions. It happens that the exponent or the base of the degree itself are expressions. In this case, the value of these expressions is first calculated, and then the value of the degree.
Example 7: The value of a numeric expression
Let's find the value of the expression 2 3 · 4 - 10 + 16 1 - 1 2 3, 5 - 2 · 1 4.
Let's start calculating in order.
2 3 4 - 10 = 2 12 - 10 = 2 2 = 4
16 · 1 - 1 2 3, 5 - 2 · 1 4 = 16 * 0, 5 3 = 16 · 1 8 = 2.
All that remains is to perform the addition operation and find out the meaning of the expression:
2 3 4 - 10 + 16 1 - 1 2 3, 5 - 2 1 4 = 4 + 2 = 6.
It is also often advisable to simplify an expression using the properties of a degree.
Example 8: The value of a numeric expression
Let's calculate the value of the following expression: 2 - 2 5 · 4 5 - 1 + 3 1 3 6 .
The exponents are again such that their exact numerical values cannot be obtained. Let's simplify the original expression to find its value.
2 - 2 5 4 5 - 1 + 3 1 3 6 = 2 - 2 5 2 2 5 - 1 + 3 1 3 6
2 - 2 5 2 2 5 - 1 + 3 1 3 6 = 2 - 2 5 2 2 5 - 2 + 3 2 = 2 2 5 - 2 - 2 5 + 3 2
2 2 5 - 2 - 2 5 + 3 2 = 2 - 2 + 3 = 1 4 + 3 = 3 1 4
Expressions with fractions
If an expression contains fractions, then when calculating such an expression, all fractions in it must be represented in the form ordinary fractions and calculate their values.
If the numerator and denominator of a fraction contain expressions, then the values of these expressions are first calculated, and the final value of the fraction itself is written down. Arithmetic operations are performed in the standard order. Let's look at the example solution.
Example 9: The value of a numeric expression
Let's find the value of the expression containing fractions: 3, 2 2 - 3 · 7 - 2 · 3 6 ÷ 1 + 2 + 3 9 - 6 ÷ 2.
As you can see, there are three fractions in the original expression. Let's first calculate their values.
3, 2 2 = 3, 2 ÷ 2 = 1, 6
7 - 2 3 6 = 7 - 6 6 = 1 6
1 + 2 + 3 9 - 6 ÷ 2 = 1 + 2 + 3 9 - 3 = 6 6 = 1.
Let's rewrite our expression and calculate its value:
1, 6 - 3 1 6 ÷ 1 = 1, 6 - 0, 5 ÷ 1 = 1, 1
Often when finding the meaning of expressions, it is convenient to reduce fractions. Exists unspoken rule: before finding its value, it is best to simplify any expression to the maximum, reducing all calculations to the simplest cases.
Example 10: The value of a numeric expression
Let's calculate the expression 2 5 - 1 - 2 5 - 7 4 - 3.
We cannot completely extract the root of five, but we can simplify the original expression through transformations.
2 5 - 1 = 2 5 + 1 5 - 1 5 + 1 = 2 5 + 1 5 - 1 = 2 5 + 2 4
The original expression takes the form:
2 5 - 1 - 2 5 - 7 4 - 3 = 2 5 + 2 4 - 2 5 - 7 4 - 3 .
Let's calculate the value of this expression:
2 5 + 2 4 - 2 5 - 7 4 - 3 = 2 5 + 2 - 2 5 + 7 4 - 3 = 9 4 - 3 = - 3 4 .
Expressions with logarithms
When logarithms are present in an expression, their value is calculated from the beginning, if possible. For example, in the expression log 2 4 + 2 · 4, you can immediately write down the value of this logarithm instead of log 2 4, and then perform all the actions. We get: log 2 4 + 2 4 = 2 + 2 4 = 2 + 8 = 10.
Under the sign of the logarithm itself and at its base there can also be numeric expressions. In this case, the first thing to do is find their meanings. Let's take the expression log 5 - 6 ÷ 3 5 2 + 2 + 7. We have:
log 5 - 6 ÷ 3 5 2 + 2 + 7 = log 3 27 + 7 = 3 + 7 = 10.
If it is impossible to calculate the exact value of the logarithm, simplifying the expression helps to find its value.
Example 11: The value of a numeric expression
Let's find the value of the expression log 2 log 2 256 + log 6 2 + log 6 3 + log 5 729 log 0, 2 27.
log 2 log 2 256 = log 2 8 = 3 .
By the property of logarithms:
log 6 2 + log 6 3 = log 6 (2 3) = log 6 6 = 1.
Using the properties of logarithms again, for the last fraction in the expression we get:
log 5 729 log 0, 2 27 = log 5 729 log 1 5 27 = log 5 729 - log 5 27 = - log 27 729 = - log 27 27 2 = - 2.
Now you can proceed to calculating the value of the original expression.
log 2 log 2 256 + log 6 2 + log 6 3 + log 5 729 log 0, 2 27 = 3 + 1 + - 2 = 2.
Expressions with trigonometric functions
It happens that the expression contains the trigonometric functions of sine, cosine, tangent and cotangent, as well as their inverse functions. The value is calculated from before all other arithmetic operations are performed. Otherwise, the expression is simplified.
Example 12: The value of a numeric expression
Find the value of the expression: t g 2 4 π 3 - sin - 5 π 2 + cosπ.
First, we calculate the values of the trigonometric functions included in the expression.
sin - 5 π 2 = - 1
We substitute the values into the expression and calculate its value:
t g 2 4 π 3 - sin - 5 π 2 + cosπ = 3 2 - (- 1) + (- 1) = 3 + 1 - 1 = 3.
The expression value has been found.
Often in order to find the meaning of an expression with trigonometric functions, it must first be converted. Let's explain with an example.
Example 13: The value of a numeric expression
We need to find the value of the expression cos 2 π 8 - sin 2 π 8 cos 5 π 36 cos π 9 - sin 5 π 36 sin π 9 - 1.
For conversion we will use trigonometric formulas cosine double angle and the cosine of the sum.
cos 2 π 8 - sin 2 π 8 cos 5 π 36 cos π 9 - sin 5 π 36 sin π 9 - 1 = cos 2 π 8 cos 5 π 36 + π 9 - 1 = cos π 4 cos π 4 - 1 = 1 - 1 = 0.
General case of a numeric expression
IN general case trigonometric expression may contain all the elements described above: brackets, powers, roots, logarithms, functions. Let us formulate a general rule for finding the meanings of such expressions.
How to find the value of an expression
- Roots, powers, logarithms, etc. are replaced by their values.
- The actions in parentheses are performed.
- The remaining actions are performed in order from left to right. First - multiplication and division, then - addition and subtraction.
Let's look at an example.
Example 14: The value of a numeric expression
Let's calculate the value of the expression - 2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 ln e 2 + 1 + 3 9.
The expression is quite complex and cumbersome. It was not by chance that we chose just such an example, having tried to fit into it all the cases described above. How to find the meaning of such an expression?
It is known that when calculating the value of a complex fractional form, first the values of the numerator and denominator of the fraction are found separately, respectively. We will sequentially transform and simplify this expression.
First of all, let's calculate the value of the radical expression 2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3. To do this, you need to find the value of the sine and the expression that is the argument of the trigonometric function.
π 6 + 2 2 π 5 + 3 π 5 = π 6 + 2 2 π + 3 π 5 = π 6 + 2 5 π 5 = π 6 + 2 π
Now you can find out the value of the sine:
sin π 6 + 2 2 π 5 + 3 π 5 = sin π 6 + 2 π = sin π 6 = 1 2.
We calculate the value of the radical expression:
2 sin π 6 + 2 2 π 5 + 3 π 5 + 3 = 2 1 2 + 3 = 4
2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 = 4 = 2.
With the denominator of the fraction everything is simpler:
Now we can write the value of the whole fraction:
2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 ln e 2 = 2 2 = 1 .
Taking this into account, we write the entire expression:
1 + 1 + 3 9 = - 1 + 1 + 3 3 = - 1 + 1 + 27 = 27 .
Final result:
2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 ln e 2 + 1 + 3 9 = 27.
In this case we were able to calculate exact values roots, logarithms, sines, etc. If this is not possible, you can try to get rid of them through mathematical transformations.
Calculating expression values using rational methods
Numeric values must be calculated consistently and accurately. This process can be streamlined and accelerated using various properties actions with numbers. For example, it is known that a product is equal to zero if at least one of the factors is equal to zero. Taking this property into account, we can immediately say that the expression 2 386 + 5 + 589 4 1 - sin 3 π 4 0 is equal to zero. At the same time, it is not at all necessary to perform the actions in the order described in the article above.
It is also convenient to use the subtraction property equal numbers. Without performing any actions, you can order that the value of the expression 56 + 8 - 3, 789 ln e 2 - 56 + 8 - 3, 789 ln e 2 is also zero.
Another technique to speed up the process is the use of identity transformations such as grouping terms and factors and subtracting common multiplier out of brackets. A rational approach to calculating expressions with fractions is to reduce the same expressions in the numerator and denominator.
For example, take the expression 2 3 - 1 5 + 3 289 3 4 3 2 3 - 1 5 + 3 289 3 4. Without performing the operations in parentheses, but by reducing the fraction, we can say that the value of the expression is 1 3 .
Finding the values of expressions with variables
Meaning literal expression and expressions with variables are found for specific given values of letters and variables.
Finding the values of expressions with variables
To find the value of a literal expression and an expression with variables, you need to substitute set values letters and variables, and then calculate the value of the resulting numerical expression.
Example 15: Value of an Expression with Variables
Calculate the value of the expression 0, 5 x - y given x = 2, 4 and y = 5.
We substitute the values of the variables into the expression and calculate:
0.5 x - y = 0.5 2.4 - 5 = 1.2 - 5 = - 3.8.
Sometimes you can transform an expression in such a way that you get its value regardless of the values of the letters and variables included in it. To do this, you need to get rid of letters and variables in the expression, if possible, using identity transformations, properties of arithmetic operations and all possible other methods.
For example, the expression x + 3 - x obviously has the value 3, and to calculate this value it is not necessary to know the value of the variable x. The value of this expression is equal to three for all values of the variable x from its range of permissible values.
Another example. The value of the expression x x is equal to one for all positive x's.
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