The rule for opening parentheses during a product. How to use simple parentheses
That part of the equation is the expression in parentheses. To open parentheses, look at the sign in front of the parentheses. If there is a plus sign, opening the parentheses in the expression will not change anything: just remove the parentheses. If there is a minus sign, when opening the brackets, you must change all the signs that were originally in the brackets to the opposite ones. For example, -(2x-3)=-2x+3.
Multiplying two parentheses.
If the equation contains the product of two brackets, expand the brackets according to the standard rule. Each term in the first bracket is multiplied with each term in the second bracket. The resulting numbers are summed up. In this case, the product of two “pluses” or two “minuses” gives the term a “plus” sign, and if the factors have different signs, then receives a minus sign.
Let's consider.
(5x+1)(3x-4)=5x*3x-5x*4+1*3x-1*4=15x^2-20x+3x-4=15x^2-17x-4.
By opening parentheses, sometimes raising an expression to . The formulas for squaring and cubed must be known by heart and remembered.
(a+b)^2=a^2+2ab+b^2
(a-b)^2=a^2-2ab+b^2
(a+b)^3=a^3+3a^2*b+3ab^2+b^3
(a-b)^3=a^3-3a^2*b+3ab^2-b^3
Formulas for constructing expressions greater than three can be done using Pascal's triangle.
Sources:
- parenthesis expansion formula
Mathematical operations enclosed in parentheses can contain variables and expressions to varying degrees complexity. To multiply such expressions, you will have to look for a solution in general view, opening the brackets and simplifying the result. If the brackets contain operations without variables, only with numerical values, then it is not necessary to open the brackets, since if you have a computer, its user has access to very significant computing resources - it is easier to use them than to simplify the expression.
Instructions
Multiply sequentially each (or minuend with ) contained in one bracket by the contents of all other brackets if you want to get the result in general form. For example, let the original expression be written like this: (5+x)∗(6-x)∗(x+2). Then sequential multiplication (that is, opening the parentheses) will give the following result: (5+x)∗(6-x)∗(x+2) = (5∗6-5∗x)∗(5∗x+5∗2) + (6∗x-x∗x)∗(x∗x+2∗x) = (5∗6∗5∗x+5∗6∗5∗2) - (5∗x∗5∗x+5∗ x∗5∗2) + (6∗x∗x∗x+6∗x∗2∗x) - (x∗x∗x∗x+x∗x∗2∗x) = 5∗6∗5∗x + 5∗6∗5∗2 - 5∗x∗5∗x - 5∗x∗5∗2 + 6∗x∗x∗x + 6∗x∗2∗x - x∗x∗x∗x - x ∗x∗2∗x = 150∗x + 300 - 25∗x² - 50∗x + 6∗x³ + 12∗x² - x∗x³ - 2∗x³.
Simplify the result by shortening the expressions. For example, the expression obtained in the previous step can be simplified as follows: 150∗x + 300 - 25∗x² - 50∗x + 6∗x³ + 12∗x² - x∗x³ - 2∗x³ = 100∗x + 300 - 13∗ x² - 8∗x³ - x∗x³.
Use a calculator if you need to multiply x equals 4.75, that is (5+4.75)∗(6-4.75)∗(4.75+2). To calculate this value, go to the Google or Nigma search engine website and enter the expression in the query field in its original form (5+4.75)*(6-4.75)*(4.75+2). Google will show 82.265625 immediately, without clicking a button, but Nigma needs to send data to the server with a click of a button.
In almost any text you can find parentheses and dashes. But users do not always format them correctly. For example, it is not uncommon to see a dash without one or two spaces, where the text is stuck to the character. The same applies to parentheses, the use of which is inappropriate or without taking into account the writing rules overloads the text. This article discusses the issues of writing parentheses and dashes in accordance with generally accepted rules.
Rules for writing parentheses
When writing parentheses, follow the same rules as for quotation marks. For example, two parentheses are not placed in a row.
There are several common cases where parentheses are used:
Individual words, groups of words and entire sentences that are not directly related to the main idea expressed by the author. Phrases spoken casually when the author does not draw the reader’s attention to them. Expressions in brackets fall outside the syntactic structure of the sentence.
Example: " And although I myself understand that when she pulls my hair, she does it only out of pity in her heart (for, I repeat without embarrassment, she pulls my hair, young man, he confirmed with great dignity, hearing the giggle again) , but, God, what if she had just once... But no! No! all this is in vain, and there is nothing to say! there is nothing to say!.. for more than once the desired thing has already happened, and more than once they have felt sorry for me, but... this is already my trait, and I am a born beast!” (F.M. Dostoevsky, “Crime and Punishment”)
Brief remarks to clarify a particular word or phrase in a sentence are placed in parentheses.
Example: " A normal, reassuring chatter ensued, when along with sincere sympathy (we all belong here, and we are all, in general, good people) There is also a hint of mocking relief. Not me! I didn’t do this stupid thing, it was clear on their faces."(S. Lukyanenko, "Shadows of Dreams")
Example: " I asked a tipsy yogi
(He ate razors and ate nails like sausage):
“Listen, friend, open up to me - by God,
I’ll take the secret with me to the grave!»
(V. Vysotsky, “Song about Yogis”)
References to formulas and illustrations are surrounded by parentheses, for example (Fig. 2), (diag. 3, page 184) , « Formula (1) is a consequence of the Pythagorean theorem. Formulas (2) And (3) are obtained from the formula (1) . » and sources of information (literature, publications) in square brackets, for example: , , etc.
Remarks are included in brackets, shining example– scenarios where the verbal embodiment is indicated in the stage directions continuous action, For example:
« Will laughs.
SKYLAR (continues)
How do you do it? I don’t... I mean, even the most smart people, whom I know, we have a couple at Harvard, we have to study - a lot. It's complicated.
(pause)
Look, Will, if you don't want to tell me...»
(Script for the film “Good Will Hunting”
Direct brackets are also used when adding unfinished words in author's papers.
Numbering in the text is written using brackets in the following format:
1)
A)
*)
Footnote signs (callouts) are designed in a similar way.
Rules for writing dashes
The dash is a punctuation mark; when writing before and after the dash, a space is always written.
There are a few exceptions where a dash is written without both or one space:
When a paragraph begins with a dash, a space is placed only after.
when a dash is placed between two numbers, acting as a hyphen. For example: " every day our site receives 3000 visitors -
3500 visitors».
For example: " - Oh... Uh... –
The dumbfounded Page could only mumble."(Philip K. Dick, "Minority Report")
Most punctuation marks, including comma, question mark, exclamation marks are placed before the dash. Example: " Central mountainous region in which the Pindus Mountains are located , - the most sparsely populated. The highest point in Greece, Mount Olympus (2917 m), is located in this region. Central Greece is the most populated region."(Eclopedical reference book "The whole world. Countries")
The dash is used in several cases:
- as a punctuation mark;
- as a connector of a pair of limiting numbers, for example: 80-90%
;
- as a mathematical minus sign;
- as a separator symbol or symbol from explanatory text, for example, when a decoding of the symbols included in the formula is given, or an explanation is given for the illustration;
- as a hyphenation sign, in this case the dash is written together with the non-hyphenated part of the word and should not be repeated at the beginning of the next line;
- like a connecting line or hyphen.
In this article we will consider in detail the basic rules of such important topic mathematics course, like opening parentheses. You need to know the rules for opening brackets in order to correctly solve equations in which they are used.
How to open parentheses correctly when adding
Expand the brackets preceded by the “+” sign
This is the simplest case, because if there is an addition sign in front of the brackets, the signs inside them do not change when the brackets are opened. Example:
(9 + 3) + (1 - 6 + 9) = 9 + 3 + 1 - 6 + 9 = 16.
How to expand parentheses preceded by a "-" sign
In this case, you need to rewrite all terms without brackets, but at the same time change all the signs inside them to the opposite ones. The signs change only for terms from those brackets that were preceded by the sign “-”. Example:
(9 + 3) - (1 - 6 + 9) = 9 + 3 - 1 + 6 - 9 = 8.
How to open parentheses when multiplying
Before the brackets there is a multiplier number
In this case, you need to multiply each term by a factor and open the brackets without changing the signs. If the multiplier has a “-” sign, then during multiplication the signs of the terms are reversed. Example:
3 * (1 - 6 + 9) = 3 * 1 - 3 * 6 + 3 * 9 = 3 - 18 + 27 = 12.
How to open two parentheses with a multiplication sign between them
In this case, you need to multiply each term from the first brackets with each term from the second brackets and then add the results. Example:
(9 + 3) * (1 - 6 + 9) = 9 * 1 + 9 * (- 6) + 9 * 9 + 3 * 1 + 3 * (- 6) + 3 * 9 = 9 - 54 + 81 + 3 - 18 + 27 = 48.
How to open parentheses in a square
If the sum or difference of two terms is squared, the brackets should be opened according to the following formula:
(x + y)^2 = x^2 + 2 * x * y + y^2.
In the case of a minus inside the brackets, the formula does not change. Example:
(9 + 3) ^ 2 = 9 ^ 2 + 2 * 9 * 3 + 3 ^ 2 = 144.
How to expand parentheses to another degree
If the sum or difference of terms is raised, for example, to the 3rd or 4th power, then you just need to break the power of the bracket into “squares”. The powers of identical factors are added, and when dividing, the power of the divisor is subtracted from the power of the dividend. Example:
(9 + 3) ^ 3 = ((9 + 3) ^ 2) * (9 + 3) = (9 ^ 2 + 2 * 9 * 3 + 3 ^ 2) * 12 = 1728.
How to open 3 brackets
There are equations in which 3 brackets are multiplied at once. In this case, you must first multiply the terms of the first two brackets together, and then multiply the sum of this multiplication by the terms of the third bracket. Example:
(1 + 2) * (3 + 4) * (5 - 6) = (3 + 4 + 6 + 8) * (5 - 6) = - 21.
These rules for opening parentheses apply equally to solving both linear and trigonometric equations.
Everywhere. Everywhere and wherever you look, you can see these constructions:
These “constructions” cause mixed reactions among literate people. At least like “is this really correct?”
In general, I personally cannot understand where the “fashion” of not closing outer quotes came from. The first and only analogy that comes to this is the analogy with parentheses. No one doubts that two brackets in a row is normal. For example: “Pay for the entire circulation (200 pieces (of which 100 are defective)).” But someone doubted the normality of putting two quotation marks in a row (I wonder who was first?) ... And now everyone has become completely clear conscience to produce structures such as Firm Pupkov and Co. LLC.
But even if you have never seen the rule in your life, which will be discussed below, then the only logical option (using the example of brackets) would be the following: LLC Firm Pupkov and Co.
So, the rule itself:
If at the beginning or at the end of a quotation (the same applies to direct speech) there are internal and external quotation marks, then they should differ from each other in design (the so-called “herringbones” and “petals”), and the external quotation marks should not be omitted, for example: C The sides of the steamer radioed: “Leningrad has entered the tropics and is continuing on its course.” About Zhukovsky, Belinsky writes: “Contemporaries of Zhukovsky’s youth looked at him primarily as an author of ballads, and in one of his letters Batyushkov called him a “balladeer.”
© Rules of Russian spelling and punctuation. - Tula: Autograph, 1995. - 192 p.
Accordingly... if you don’t have the opportunity to type “herringbone” quotes, then what can you do, you’ll have to use such “” icons. However, the inability (or unwillingness) to use Russian quotation marks is by no means a reason why you can not close external quotation marks.Thus, the inaccuracy of the design of LLC "Firm Pupkov and Co" seems to have been sorted out. There are also designs of the type LLC Firm "Pupkov and Co".
It is completely clear from the rule that such constructions are also illiterate... (Correct: LLC "Firm "Pupkov and Co""However!
The Publisher's and Author's Guide by A.E. Milchin (2004 edition) states that two design options can be used in such cases. The use of “herringbones” and “legs” and (in the absence of technical means) the use of only “herringbones”: two opening ones and one closing one.
The directory is “fresh” and personally, I immediately have 2 questions here. Firstly, with what joy can one use one closing quotation mark (well, this is illogical, see above), and secondly, the phrase “in the absence of technical means” especially attracts attention. How is this, excuse me? Now open Notepad and type “only Christmas trees: two opening and one closing.” There are no such symbols on the keyboard. I can’t print “herringbone”... The combination Shift + 2 produces the sign " (which, as you know, is not a quotation mark). Now open Microsoft Word and press Shift + 2 again. The program will correct " to " (or "). Well, it turns out that the rule that existed for decades was taken and rewritten under Microsoft Word? Like, since the Word from the "Firm "Pupkov and Co" makes "Firm "Pupkov and Co", then let this now be acceptable and correct???
It seems so. And if this is so, then there is every reason to doubt the correctness of such an innovation.Yes, and one more clarification... about the very “lack of technical means.” The fact is that on any computer with Windows there are always “technical means” for entering both “Christmas trees” and “legs”, so this new “rule” (for me it is in quotes) is incorrect from the very beginning!
All special characters in a font can be easily typed by knowing the corresponding number of that character. Just hold down Alt and type on the NumLock keyboard (NumLock is pressed, the indicator light is on) the corresponding symbol number:
„ Alt + 0132 (left “foot”)
“ Alt + 0147 (right foot)
« Alt + 0171 (left herringbone)
» Alt + 0187 (right herringbone)
A+(b + c) can be written without parentheses: a+(b + c)=a + b + c. This operation is called opening parentheses.
Example 1. Let's open the brackets in the expression a + (- b + c).
Solution. a + (-b+c) = a + ((-b) + c)=a + (-b) + c = a-b + c.
If there is a “+” sign in front of the brackets, then you can omit the brackets and this “+” sign while maintaining the signs of the terms in the brackets. If the first term in brackets is written without a sign, then it must be written with a “+” sign.
Example 2. Let's find the value of the expression -2.87+ (2.87-7.639).
Solution. Opening the brackets, we get - 2.87 + (2.87 - 7.639) = - - 2.87 + 2.87 - 7.639 = 0 - 7.639 = - 7.639.
To find the value of the expression - (- 9 + 5), you need to add numbers-9 and 5 and find the number opposite to the resulting sum: -(- 9 + 5)= -(- 4) = 4.
The same value can be obtained in another way: first write down the numbers opposite to these terms (i.e. change their signs), and then add: 9 + (- 5) = 4. Thus, -(- 9 + 5) = 9 - 5 = 4.
To write a sum opposite to the sum of several terms, you need to change the signs of these terms.
This means - (a + b) = - a - b.
Example 3. Let's find the value of the expression 16 - (10 -18 + 12).
Solution. 16-(10 -18 + 12) = 16 + (-(10 -18 + 12)) = = 16 + (-10 +18-12) = 16-10 +18-12 = 12.
To open brackets preceded by a “-” sign, you need to replace this sign with “+”, changing the signs of all terms in the brackets to the opposite, and then open the brackets.
Example 4. Let's find the value of the expression 9.36-(9.36 - 5.48).
Solution. 9.36 - (9.36 - 5.48) = 9.36 + (- 9.36 + 5.48) = = 9.36 - 9.36 + 5.48 = 0 -f 5.48 = 5 ,48.
Expanding parentheses and applying commutative and associative properties addition allow you to simplify calculations.
Example 5. Let's find the value of the expression (-4-20)+(6+13)-(7-8)-5.
Solution. First, let’s open the brackets, and then find separately the sum of all positive ones and separately the sum of all negative numbers and finally, add up the results:
(- 4 - 20)+(6+ 13)-(7 - 8) - 5 = -4-20 + 6 + 13-7 + 8-5 = = (6 + 13 + 8)+(- 4 - 20 - 7 - 5)= 27-36=-9.
Example 6. Let's find the value of the expression
Solution. First, let’s imagine each term as the sum of their integer and fractional parts, then open the brackets, then add the integers and separately fractional parts and finally add up the results:
How do you open parentheses preceded by a “+” sign? How can you find the value of an expression that is the opposite of the sum of several numbers? How to expand parentheses preceded by a “-” sign?
1218. Open the brackets:
a) 3.4+(2.6+ 8.3); c) m+(n-k);
b) 4.57+(2.6 - 4.57); d) c+(-a + b).
1219. Find the meaning of the expression:
1220. Open the brackets:
a) 85+(7.8+ 98); d) -(80-16) + 84; g) a-(b-k-n);
b) (4.7 -17)+7.5; e) -a + (m-2.6); h) -(a-b + c);
c) 64-(90 + 100); e) c+(- a-b); i) (m-n)-(p-k).
1221. Open the brackets and find the meaning of the expression:
1222. Simplify the expression:
1223. Write amount two expressions and simplify it:
a) - 4 - m and m + 6.4; d) a+b and p - b
b) 1.1+a and -26-a; e) - m + n and -k - n;
c) a + 13 and -13 + b; e)m - n and n - m.
1224. Write the difference of two expressions and simplify it:
1226. Use the equation to solve the problem:
a) There are 42 books on one shelf, and 34 on the other. Several books were removed from the second shelf, and as many books were taken from the first shelf as were left on the second. After that, there were 12 books left on the first shelf. How many books were removed from the second shelf?
b) There are 42 students in the first grade, 3 students less in the second than in the third. How many students are there in third grade if there are 125 students in these three grades?
1227. Find the meaning of the expression:
1228. Calculate orally:
1229. Find highest value expressions:
1230. Specify 4 consecutive integers if:
a) the smaller of them is -12; c) the smaller of them is n;
b) the largest of them is -18; d) the greater of them is equal to k.
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