Let's look at this algorithm using an example. We'll find
1st step. We divide the number under the root into two-digit faces (from right to left):
2nd step. We extract Square root from the first face, i.e. from the number 65, we get the number 8. Under the first face we write the square of the number 8 and subtract. We assign the second face (59) to the remainder:
(number 159 is the first remainder).
3rd step. We double the found root and write the result on the left:
4th step. We separate one digit on the right in the remainder (159), and on the left we get the number of tens (it is equal to 15). Then we divide 15 by double the first digit of the root, i.e. by 16, since 15 is not divisible by 16, the quotient results in zero, which we write as the second digit of the root. So, in the quotient we got the number 80, which we double again, and remove the next edge
(the number 15,901 is the second remainder).
5th step. In the second remainder we separate one digit from the right and divide the resulting number 1590 by 160. We write the result (number 9) as the third digit of the root and add it to the number 160. We multiply the resulting number 1609 by 9 and find the next remainder (1420):
Subsequently, actions are performed in the sequence specified in the algorithm (the root can be extracted with the required degree of accuracy).
Comment. If the radical expression is a decimal fraction, then its whole part is divided into edges of two digits from right to left, the fractional part - two digits from left to right, and the root is extracted according to the specified algorithm.
DIDACTIC MATERIAL
1. Take the square root of the number: a) 32; b) 32.45; c) 249.5; d) 0.9511.
What is a square root?
Attention!
There are additional
materials in Special Section 555.
For those who are very "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 - there is also subtraction. There is multiplication - there is also division. There is squaring... So there is also taking the square root! That's all. This action ( square root) in mathematics is indicated by this icon:
The icon itself is called a beautiful word "radical".
How to extract the root? It's better to look at examples.
What is the square root of 9? What number squared will give us 9? 3 squared gives us 9! Those:
But what is the square root of zero? No problem! What number squared does zero make? Yes, it gives zero! Means:
Got it, what is square root? Then we consider examples:
Answers (in disarray): 6; 1; 4; 9; 5.
Decided? Really, how much easier is that?!
But... What does a person do when he sees some task with roots?
A person begins to feel sad... He does not believe in the simplicity and lightness of his roots. Although he seems to know what is square root...
This is because the person ignored several important points when studying the roots. Then these fads take cruel revenge on tests and exams...
Point one. You need to recognize the roots by sight!
What is the square root of 49? Seven? Right! How did you know it was seven? Squared seven and got 49? Right! 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 have missed...
This is the difficulty root extraction. Square any number without special problems. Multiply a number by itself with a column - that's all. But for root extraction There is no such simple and fail-safe technology. We have to pick up answer and check if it is correct by squaring it.
This one is complicated 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 four 6 times, do you? The answer 24 immediately comes up. Although, not everyone gets 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 recite 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 be a great help in solving examples. The second is to solve more examples. This will greatly help you remember the table of squares.
And no calculators! For testing purposes only. Otherwise, you will slow down mercilessly during the exam...
So, what is square root And How extract roots- I think it’s clear. Now let's find out WHAT we can extract them from.
Point two. Root, I don't know you!
What numbers can you take square roots from? Yes, almost any of them. It's easier to understand what it's from it is forbidden extract them.
Let's try to calculate this root:
To do this, we need to choose a number that squared will give us -4. We select.
What, it doesn't fit? 2 2 gives +4. (-2) 2 gives again +4! That's it... There are no numbers that, when squared, will give us a negative number! Although I know these numbers. But I won’t tell you). Go to college and you will find out for yourself.
The same story will happen with any negative number. Hence the conclusion:
An expression in which there is a negative number under the square root sign - doesn't make sense! This is a forbidden operation. It is as forbidden as dividing by zero. Remember this fact firmly! Or in other words:
You cannot extract square roots from negative numbers!
But of all the others, it’s possible. For example, it is quite possible to calculate
At first glance, this is very difficult. Selecting fractions and squaring them... Don't worry. When we understand the properties of 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, gives us two. Only this number is completely uneven... Here it is:
What’s interesting is that 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 an example, you end up with something that cannot be extracted, like:
then we leave it like that. This will be the answer.
You need to clearly understand what the icons mean
Of course, if the root of the number is taken smooth, you must do this. The answer to the task is in the form, for example
Quite a complete answer.
And, of course, you need to know the approximate values from memory:
This knowledge greatly helps to assess the situation in complex tasks.
Point three. The most cunning.
The main confusion in working with roots is caused by this point. It is he who gives uncertainty to own strength... Let's deal with this issue properly!
First, let's take the square root of four of them again. Have I already bothered you with this root?) Never mind, now it will be interesting!
What number does 4 square? Well, two, two - I hear dissatisfied answers...
Right. Two. But also minus two will give 4 squared... Meanwhile, the answer
correct and the answer
gross 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 mathematics course, it is customary to consider square roots only non-negative numbers! That is, zero and all are positive. Even a special term was invented: from the number A- This non-negative number whose square is A. Negative results when extracting an arithmetic square root are simply discarded. At school, everything is square roots - arithmetic. Although this is not particularly mentioned.
Okay, that's understandable. It's even better not to bother with negative results... This is not yet confusion.
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 (absolutely correct, by the way) is just an abbreviated version two answers:
Stop, stop! Just above 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 (just for understanding!) like this:
The parentheses do not change the essence of the answer. I just separated it with brackets signs from root. Now you can clearly see that the root itself (in brackets) is still a non-negative number! And the signs are result of solving the equation. After all, when solving any equation we must write All Xs that, when substituted into the original equation, will give the correct result. The root of five (positive!) with both a plus and a minus fits into our equation.
Like this. If you just take the square root from anything, you Always you get one non-negative result. For example:
Because it - arithmetic square root.
But if you decide something quadratic equation, type:
That Always it turns out two answer (with plus and minus):
Because this is the solution to the equation.
Hope, what is square root You've got your points clear. Now it remains to find out what can be done with the roots, what their properties are. And what are the points and pitfalls... sorry, stones!)
All this is in the following 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. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
In mathematics, the question of how to extract a root is considered relatively simple. 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 row 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.
Root of a number: calculation rules and examples
So, we squared the number 2, that is, multiplied it by itself and got 4. How to extract the root of the number 4? Let's say right away that the roots can be square, cubic and any degree to infinity.
Root degree – always natural number, that is, it is impossible to solve such an equation: a 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 extract 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 quite 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 was said above, 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 required number was initially multiplied by itself not once, but twice. That is, let's say we took the following example:
- 3x3x3=27
- Naturally, the cube root of 27 is three:
- Root 3 of 27 = 3
Let's say 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 roots through practice, by solving many examples and memorizing tables of squares and cubes of small numbers. In the future, this will greatly facilitate and reduce the time required to solve equations. Although, it should be noted that sometimes it is necessary to extract the root of such large number that finding the correct squared number would be very difficult, if possible at all. A regular calculator will come to the rescue in extracting the square root. How to extract the root on a calculator? Very simply enter the number from which you want to find the result. Now take a close look at the calculator buttons. Even the simplest of them has a key with a root icon. By clicking on it, you will immediately get the finished result.
Not every number can have a whole root; consider the following example:
Root of 1859 = 43.116122…
You can simultaneously try to solve this example on a calculator. As you can see, the resulting number is not an integer; moreover, the set of digits after the decimal point is not finite. Special engineering calculators can give a more accurate result, 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 was squared to obtain it is not an integer.
If you need to extract the third root on a simple calculator, then you need to double-click on the button with the root sign. For example, take the number 1859 used above and take 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 get approximately 1859. Thus, extracting roots from numbers is not difficult, you just need to remember the above algorithms.
Bibliographic description: Pryostanovo S. M., Lysogorova L. V. Methods for extracting the square root // Young scientist. 2017. No. 2.2. P. 76-77..02.2019).
Keywords : square root, square root extraction.
In mathematics lessons, I became acquainted with the concept of a square root, and the operation of extracting a square root. I became interested in whether extracting the square root is possible only 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 solving equations, the method of discarding a complete square, Newton's method, the geometric method, the graphical method (, ), the guessing method, the method of odd number deductions.
Consider the following methods:
Let's factor it into prime factors using the divisibility criteria 27225=5*5*3*3*11*11. Thus
- TO Canadian method. This quick method was discovered by young scientists at one of Canada's leading universities in the 20th century. Its accuracy is no more than two to three decimal places.
where x is the number from which the root must be extracted, c is the number of the nearest square), for example:
=5,92
- In a column. This method allows you to find the approximate value of the root of any real number with any predetermined accuracy. The disadvantages of this method include the increasing complexity of the calculation as the number of digits found increases. To manually extract the root, a notation similar to long division is used
Square Root Algorithm
1. We divide the fractional part and the integer part separately from the comma on the verge of two digits 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 on the first side.
3. Find the difference between the number on the first face and the square of the selected first number.
4. We add the next edge to the resulting difference, the resulting number will be divisible. Let's educate 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 entire divisor does not exceed the dividend. We write down the selected number as an answer.
5. We take the next edge to the resulting difference and perform the actions according to the algorithm. If this face turns out to be a face of a fractional part, then we put a comma in the answer. (Fig. 1.)
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Using this method, you can extract numbers with different precisions, for example, up to thousandths. (Fig.2)
Considering various ways 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 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 square roots and provides examples of extracting roots.
In the preface to his first edition, “In the Kingdom of Ingenuity” (1908), E. I. Ignatiev writes: “... intellectual initiative, quick wit and “ingenuity” cannot be “drilled into” or “put into” anyone’s head. The results are reliable only when the introduction to the field of mathematical knowledge is made in an easy and nice shape, on objects and examples of ordinary and everyday situations, selected with appropriate wit and entertainment.”
In the preface to the 1911 edition “The Role of Memory in Mathematics” E.I. Ignatiev writes “... in mathematics it is not the formulas that should be remembered, but the process of thinking.”
To extract the square root, there are tables of squares for two-digit numbers; you can factor the number into prime factors and extract the square root of the product. A table of squares is sometimes not enough; extracting the root by factoring is a time-consuming task, which also does not always lead to the desired result. Try taking the square root of 209764? Factoring into prime factors gives the product 2*2*52441. By trial and error, selection - this, of course, can be done if you are sure that this is an integer. The method I want to propose allows you to take the square root in any case.
Once upon a time at the institute (Perm State Pedagogical Institute) we were introduced to this method, which I now want to talk about. I never wondered whether this method had a proof, so now I had to deduce some of the proof myself.
The basis of this method is the composition of the number =.
=&, i.e. & 2 =596334.
1. Divide the number (5963364) into pairs from right to left (5`96`33`64)
2. Extract the square root of the first group on the left ( - number 2). This is how we get the first digit of &.
3. Find the square of the first digit (2 2 =4).
4. Find the difference between the first group and the square of the first digit (5-4=1).
5. We take down the next two digits (we get the number 196).
6. Double the first digit we found and write it on the left behind the line (2*2=4).
7. Now we need to find the second digit of the number &: double the first digit we found becomes the tens digit of the number, which when multiplied by the number of units, you need to get a number less than 196 (this is the number 4, 44*4=176). 4 is the second digit of &.
8. Find the difference (196-176=20).
9. We demolish the next group (we get the number 2033).
10. Double the number 24, we get 48.
There are 11.48 tens in a number, when multiplied by the number of ones, we should get a number less than 2033 (484*4=1936). The ones digit we found (4) is the third digit of the number &.
I have given the proof for the following cases:
1. Extracting the square root of a three-digit number;
2. Extracting the square root of a four-digit number.
Approximate methods for extracting square roots (without using a calculator).
1. The ancient Babylonians used the following method to find the approximate value of the square root of their number x. They represented the number x as the sum a 2 + b, where a 2 is the exact square of the natural number a (a 2 ? x) closest to the number x, and used the formula 
. (1)
Using formula (1), we extract the square root, for example, from the number 28:
The result of extracting the root of 28 using MK is 5.2915026.
As we see, the Babylonian method gives a good approximation to exact value root
2. Isaac Newton developed a method for taking square roots that dates back to Heron of Alexandria (circa 100 AD). This method (known as Newton's method) is as follows.
Let a 1- the first approximation of a number (as a 1 you can take the values of the square root of a natural number - an exact square not exceeding X) .
Next, more accurate approximation a 2 numbers
found by the formula 
.