Bibliographic description: Pryamostanov S. M., Lysogorova L. V. Methods for extracting a square root // Young scientist. - 2017. - No. 2.2. - S. 76-77..02.2019).
Keywords : square root, square root extraction.
At the lessons of mathematics, I got acquainted with the concept of a square root, and the operation of extracting a square root. I became interested in extracting the square root is only possible using a table of squares, using a calculator, or is there a way to extract it manually. I found several ways: the formula of Ancient Babylon, through the solution of equations, the method of discarding the full square, Newton's method, the geometric method, the graphic method (, ), the guessing method, the odd number subtraction method.
Consider the following methods:
Let's decompose into prime factors using the signs of divisibility 27225=5*5*3*3*11*11. Thus
- To Canadian method. This fast method was discovered by young scientists at one of Canada's leading universities in the 20th century. Its accuracy is no more than two or three decimal places.
where x is the number to take the root from, c is the number of the nearest square), for example:
=5,92
- column. This method allows you to find the approximate value of the root of any real number with any predetermined accuracy. The disadvantages of the method include the increasing complexity of the calculation with an increase in the number of digits found. To manually extract the root, a notation similar to division by a column is used.
Square Root Algorithm
1. Separately divide the fractional part and the integer part separately from the comma on the edge of two numbers in each face ( kiss part - from right to left; fractional- from left to right). It is possible that the integer part may contain one digit, and the fractional part may contain zeros.
2. Extraction starts from left to right, and we select a number whose square does not exceed the number in the first face. We square this number and write it under the number in the first face.
3. We find the difference between the number in the first face and the square of the selected first number.
4. To the resulting difference we demolish the next face, the resulting number will be divisible. We form divider. We double the first selected digit of the answer (multiply by 2), we get the number of tens of the divisor, and the number of units should be such that its product by the whole divisor does not exceed the dividend. We write down the selected number in the answer.
5. To the resulting difference, we demolish the next face and perform actions according to the algorithm. If this face turns out to be the face of the fractional part, then put a comma in the answer. (Fig. 1.)
|
|
|
In this way, you can extract numbers with different accuracy, for example, with an accuracy of thousandths. (Fig.2)
Considering the various methods of extracting the square root, we can conclude: in each specific case, you need to decide on the choice of the most effective one in order to spend less time on solving
Literature:
- Kiselev A. Elements of Algebra and Analysis. Part one.-M.-1928
Keywords: square root, square root.
Annotation: The article describes methods for extracting a square root, and provides examples of extracting roots.
In mathematics, the question of how to take a root is considered relatively easy. If we square numbers from the natural series: 1, 2, 3, 4, 5 ... n, then we get the following series of squares: 1, 4, 9, 16 ... n 2. The series of squares is infinite, and if you look closely at it, you will see that there are not very many integers in it. Why this is so will be explained a little later.
The root of the number: calculation rules and examples
So, we squared the number 2, that is, we multiplied it by itself and got 4. But how to take the root of the number 4? Let's say right away that the roots can be square, cubic, and any degree up to infinity.
The degree of the root is always a natural number, that is, it is impossible to solve such an equation: the root to the power of 3.6 of n.
Square root
Let's return to the question of how to extract the square root of 4. Since we squared the number 2, we will also extract the square root. In order to correctly take the root of 4, you just need to choose the right number that, when squared, would give the number 4. And this, of course, is 2. Look at the example:
- 2 2 =4
- Root of 4 = 2
This example is pretty simple. Let's try to extract the square root of 64. What number, when multiplied by itself, gives 64? Obviously it's 8.
- 8 2 =64
- Root of 64=8
cube root
As mentioned above, the roots are not only square, using an example we will try to explain more clearly how to extract a cube root or a root of the third degree. The principle of extracting a cube root is the same as that of a square root, the only difference is that the desired number was initially multiplied by itself not once, but twice. So, let's say we take the following example:
- 3x3x3=27
- Naturally, the cube root of the number 27 will be three:
- Root 3 of 27 = 3
Suppose you need to find the cube root of 64. To solve this equation, it is enough to find a number that, when raised to the third power, would give 64.
- 4 3 =64
- Root 3 of 64 = 4
Extract the root of a number on a calculator
Of course, it is best to learn to extract square, cube and other degrees by practice, by solving many examples and memorizing a table of squares and cubes of small numbers. In the future, this will greatly facilitate and reduce the time for solving equations. Although, it should be noted that sometimes it is required to extract the root of such a large number that it will cost a lot of work, if at all, to find the correct squared number. An ordinary calculator will come to the rescue in extracting the square root. How to take a root on a calculator? It is very simple to enter the number from which you want to find the result. Now take a close look at the calculator buttons. Even on the simplest of them, there is a key with a root icon. By clicking on it, you will immediately get the finished result.
Not every number can be taken as a whole root, consider the following example:
Root of 1859 = 43.116122…
You can try to solve this example on a calculator in parallel. As you can see, the resulting number is not an integer; moreover, the set of digits after the decimal point is not finite. A more accurate result can be given by special engineering calculators, but the full result simply does not fit on the display of ordinary ones. And if you continue the series of squares that you started earlier, you will not find the number 1859 in it, precisely because the number that you squared to get it is not an integer.
If you need to extract the root of the third degree on a simple calculator, then you need to double-click on the button with the root sign. For example, let's take the number 1859 used above and extract the cube root from it:
Root 3 of 1859 = 6.5662867…
That is, if the number 6.5662867 ... is raised to the third power, then we will get approximately 1859. Thus, extracting roots from numbers is not difficult, just remember the above algorithms.
What is a square root?
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
This concept is very simple. Natural, I would say. Mathematicians try to find a reaction for every action. There is addition and there is subtraction. There is multiplication and there is division. There is squaring ... So there is also extracting the square root! That's all. This action ( taking the square root) in mathematics is denoted by this icon:
The icon itself is called the beautiful word " radical".
How to extract the root? It is better to consider examples.
What is the square root of 9? And what number squared will give us 9? 3 squared gives us 9! Those:
What is the square root of zero? No problem! What number squared zero gives? Yes, he himself gives zero! Means:
Caught what is a square root? Then we consider examples:
Answers (in disarray): 6; one; 4; nine; 5.
Decided? Really, it's much easier!
But... What does a person do when he sees some task with roots?
A person begins to yearn ... He does not believe in the simplicity and lightness of the roots. Although he seems to know what is square root...
This is because a person has ignored several important points when studying the roots. Then these fads brutally take revenge on tests and exams ...
Point one. Roots must be recognized by sight!
What is the square root of 49? Seven? Right! How did you know there were seven? Squared seven and got 49? Correctly! Please note that extract the root out of 49, we had to do the reverse operation - square 7! And make sure we don't miss. Or they could miss...
Therein lies the difficulty root extraction. Squaring any number is possible without any problems. Multiply the number by itself in a column - and that's all. But for root extraction there is no such simple and trouble-free technology. account for pick up answer and check it for hit by squaring.
This complex creative process - choosing an answer - is greatly simplified if you remember squares of popular numbers. Like a multiplication table. If, say, you need to multiply 4 by 6 - you don’t add the four 6 times, do you? The answer immediately pops up 24. Although, not everyone has it, yes ...
For free and successful work with roots, it is enough to know the squares of numbers from 1 to 20. Moreover, there and back. Those. you should be able to easily name both, say, 11 squared and the square root of 121. To achieve this memorization, there are two ways. The first is to learn the table of squares. This will help a lot with examples. The second is to solve more examples. It's great to remember the table of squares.
And no calculators! For verification only. Otherwise, you will slow down mercilessly during the exam ...
So, what is square root And How extract roots- I think it's understandable. Now let's find out FROM WHAT you can extract them from.
Point two. Root, I don't know you!
What numbers can you take square roots from? Yes, almost any. It's easier to understand what it is forbidden extract them.
Let's try to calculate this root:
To do this, you need to pick up a number that squared will give us -4. We select.
What is not selected? 2 2 gives +4. (-2) 2 gives +4 again! That's it ... There are no numbers that, when squared, will give us a negative number! Even though I know the numbers. But I won't tell you.) Go to college and find out for yourself.
The same story will be with any negative number. Hence the conclusion:
An expression in which a negative number is under the square root sign - doesn't make sense! This is a prohibited operation. As forbidden as division by zero. Keep this fact in mind! Or, in other words:
You can't extract square roots from negative numbers!
But of all the rest - you can. For example, it is possible to calculate
At first glance, this is very difficult. Pick up fractions, but square up ... Don't worry. When we deal with the properties of the roots, such examples will be reduced to the same table of squares. Life will become easier!
Okay fractions. But we still come across expressions like:
It's OK. All the same. The square root of two is the number that, when squared, will give us a deuce. Only the number is completely uneven ... Here it is:
Interestingly, this fraction never ends... Such numbers are called irrational. In square roots, this is the most common thing. By the way, this is why expressions with roots are called irrational. It is clear that writing such an infinite fraction all the time is inconvenient. Therefore, instead of an infinite fraction, they leave it like this:
If, when solving the example, you get something that is not extractable, such as:
then we leave it like that. This will be the answer.
You need to clearly understand what is under the icons
Of course, if the root of the number is taken smooth, you must do so. The answer of the task in the form, for example
quite a complete answer.
And, of course, you need to know the approximate values from memory:
This knowledge helps a lot to assess the situation in complex tasks.
Point three. The most cunning.
The main confusion in the work with the roots is brought just by this fad. It is he who gives self-doubt ... Let's deal with this fad properly!
To begin with, we again extract the square root of their four. What, have I already got you with this root?) Nothing, now it will be interesting!
What number will give in the square of 4? Well, two, two - I hear dissatisfied answers ...
Right. Two. But also minus two will give 4 squared ... Meanwhile, the answer
correct and the answer
grossest mistake. Like this.
So what's the deal?
Indeed, (-2) 2 = 4. And under the definition of the square root of four minus two quite suitable ... This is also the square root of four.
But! In the school course of mathematics, it is customary to consider square roots only non-negative numbers! Ie zero and all positive. Even a special term was coined: from the number a- This non-negative number whose square is a. Negative results when extracting the arithmetic square root are simply discarded. At school, all square roots - arithmetic. Though it's not specifically mentioned.
Okay, that's understandable. It's even better not to mess around with negative results... It's not confusion yet.
The confusion begins when solving quadratic equations. For example, you need to solve the following equation.
The equation is simple, we write the answer (as taught):
This answer (quite correct, by the way) is just an abbreviated notation two answers:
Stop stop! A little higher I wrote that the square root is a number always non-negative! And here is one of the answers - negative! Disorder. This is the first (but not the last) problem that causes distrust of the roots ... Let's solve this problem. Let's write down the answers (purely for understanding!) like this:
The parentheses do not change the essence of the answer. I just separated with brackets signs from root. Now it is clearly seen that the root itself (in brackets) is still a non-negative number! And the signs are the result of solving the equation. After all, when solving any equation, we must write all x, which, when substituted into the original equation, will give the correct result. The root of five (positive!) is suitable for our equation with both plus and minus.
Like this. If you just take the square root from anything you always get one non-negative result. For example:
Because it - arithmetic square root.
But if you solve some quadratic equation like:
then always it turns out two answer (with plus and minus):
Because it is the solution to an equation.
Hope, what is square root you got it right with your points. Now it remains to find out what can be done with the roots, what are their properties. And what are the fads and underwater boxes ... excuse me, stones!)
All this - in the next lessons.
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. Learning - with interest!)
you can get acquainted with functions and derivatives.
Root formulas. properties of square roots.
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
In the previous lesson, we figured out what a square root is. It's time to figure out what are formulas for roots, what are root properties and what can be done about it all.
Root Formulas, Root Properties, and Rules for Actions with Roots- it's essentially the same thing. There are surprisingly few formulas for square roots. Which, of course, pleases! Rather, you can write a lot of all sorts of formulas, but only three are enough for practical and confident work with roots. Everything else flows from these three. Although many stray in the three formulas of the roots, yes ...
Let's start with the simplest. Here she is:
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. Learning - with interest!)
you can get acquainted with functions and derivatives.
Students always ask: “Why can't I use a calculator on a math exam? How to extract the square root of a number without a calculator? Let's try to answer this question.
How to extract the square root of a number without the help of a calculator?
Action square root extraction the opposite of squaring.
√81= 9 9 2 =81
If we take the square root of a positive number and square the result, we get the same number.
From small numbers that are exact squares of natural numbers, for example 1, 4, 9, 16, 25, ..., 100, square roots can be extracted verbally. Usually at school they teach a table of squares of natural numbers up to twenty. Knowing this table, it is easy to extract the square roots from the numbers 121,144, 169, 196, 225, 256, 289, 324, 361, 400. From numbers greater than 400, you can extract using the selection method using some tips. Let's try an example to consider this method.
Example: Extract the root of the number 676.
We notice that 20 2 \u003d 400, and 30 2 \u003d 900, which means 20< √676 < 900.
Exact squares of natural numbers end in 0; one; 4; 5; 6; nine.
The number 6 is given by 4 2 and 6 2 .
So, if the root is taken from 676, then it is either 24 or 26.
It remains to check: 24 2 = 576, 26 2 = 676.
Answer: √676 = 26 .
More example: √6889 .
Since 80 2 \u003d 6400, and 90 2 \u003d 8100, then 80< √6889 < 90.
The number 9 is given by 3 2 and 7 2, then √6889 is either 83 or 87.
Check: 83 2 = 6889.
Answer: √6889 = 83 .
If you find it difficult to solve by the selection method, then you can factorize the root expression.
For example, find √893025.
Let's factorize the number 893025, remember, you did it in the sixth grade.
We get: √893025 = √3 6 ∙5 2 ∙7 2 = 3 3 ∙5 ∙7 = 945.
More example: √20736. Let's factorize the number 20736:
We get √20736 = √2 8 ∙3 4 = 2 4 ∙3 2 = 144.
Of course, factoring requires knowledge of divisibility criteria and factoring skills.
And finally, there is square root rule. Let's look at this rule with an example.
Calculate √279841.
To extract the root of a multi-digit integer, we split it from right to left into faces containing 2 digits each (there may be one digit in the left extreme face). Write like this 27'98'41
To get the first digit of the root (5), we extract the square root of the largest exact square contained in the first left face (27).
Then the square of the first digit of the root (25) is subtracted from the first face and the next face (98) is attributed (demolished) to the difference.
To the left of the received number 298, they write the double digit of the root (10), divide by it the number of all tens of the previously obtained number (29/2 ≈ 2), experience the quotient (102 ∙ 2 = 204 should be no more than 298) and write (2) after the first digit of the root.
Then the resulting quotient 204 is subtracted from 298, and the next facet (41) is attributed (demolished) to the difference (94).
To the left of the resulting number 9441, they write the double product of the digits of the root (52 ∙ 2 = 104), divide by this product the number of all tens of the number 9441 (944/104 ≈ 9), experience the quotient (1049 ∙ 9 = 9441) should be 9441 and write it down (9) after the second digit of the root.
We got the answer √279841 = 529.
Similarly extract roots of decimals. Only the radical number must be divided into faces so that the comma is between the faces.
Example. Find the value √0.00956484.
Just remember that if the decimal fraction has an odd number of decimal places, the square root is not exactly extracted from it.
So, now you have seen three ways to extract the root. Choose the one that suits you best and practice. To learn how to solve problems, you need to solve them. And if you have any questions, sign up for my lessons.
site, with full or partial copying of the material, a link to the source is required.
