Fact 1.
\(\bullet\) Take some non-negative number \(a\) (ie \(a\geqslant 0\) ). Then (arithmetic) square root from the number \(a\) such a non-negative number \(b\) is called, when squaring it we get the number \(a\) : \[\sqrt a=b\quad \text(same as )\quad a=b^2\] It follows from the definition that \(a\geqslant 0, b\geqslant 0\). These restrictions are an important condition for the existence of a square root and should be remembered!
Recall that any number when squared gives a non-negative result. That is, \(100^2=10000\geqslant 0\) and \((-100)^2=10000\geqslant 0\) .
\(\bullet\) What is \(\sqrt(25)\) ? We know that \(5^2=25\) and \((-5)^2=25\) . Since by definition we have to find a non-negative number, \(-5\) is not suitable, hence \(\sqrt(25)=5\) (since \(25=5^2\) ).
Finding the value \(\sqrt a\) is called taking the square root of the number \(a\) , and the number \(a\) is called the root expression.
\(\bullet\) Based on the definition, the expressions \(\sqrt(-25)\) , \(\sqrt(-4)\) , etc. don't make sense.

Fact 2.
For quick calculations, it will be useful to learn the table of squares of natural numbers from \(1\) to \(20\) : \[\begin(array)(|ll|) \hline 1^2=1 & \quad11^2=121 \\ 2^2=4 & \quad12^2=144\\ 3^2=9 & \quad13 ^2=169\\ 4^2=16 & \quad14^2=196\\ 5^2=25 & \quad15^2=225\\ 6^2=36 & \quad16^2=256\\ 7^ 2=49 & \quad17^2=289\\ 8^2=64 & \quad18^2=324\\ 9^2=81 & \quad19^2=361\\ 10^2=100& \quad20^2= 400\\ \hline \end(array)\]

Fact 3.
What can be done with square roots?
\(\bullet\) The sum or difference of square roots is NOT EQUAL to the square root of the sum or difference, i.e. \[\sqrt a\pm\sqrt b\ne \sqrt(a\pm b)\] Thus, if you need to calculate, for example, \(\sqrt(25)+\sqrt(49)\) , then initially you must find the values ​​\(\sqrt(25)\) and \(\sqrt(49)\ ) and then add them up. Consequently, \[\sqrt(25)+\sqrt(49)=5+7=12\] If the values ​​\(\sqrt a\) or \(\sqrt b\) cannot be found when adding \(\sqrt a+\sqrt b\), then such an expression is not further converted and remains as it is. For example, in the sum \(\sqrt 2+ \sqrt (49)\) we can find \(\sqrt(49)\) - this is \(7\) , but \(\sqrt 2\) cannot be converted in any way, that's why \(\sqrt 2+\sqrt(49)=\sqrt 2+7\). Further, this expression, unfortunately, cannot be simplified in any way.\(\bullet\) The product/quotient of square roots is equal to the square root of the product/quotient, i.e. \[\sqrt a\cdot \sqrt b=\sqrt(ab)\quad \text(s)\quad \sqrt a:\sqrt b=\sqrt(a:b)\] (provided that both parts of the equalities make sense)
Example: \(\sqrt(32)\cdot \sqrt 2=\sqrt(32\cdot 2)=\sqrt(64)=8\); \(\sqrt(768):\sqrt3=\sqrt(768:3)=\sqrt(256)=16\); \(\sqrt((-25)\cdot (-64))=\sqrt(25\cdot 64)=\sqrt(25)\cdot \sqrt(64)= 5\cdot 8=40\). \(\bullet\) Using these properties, it is convenient to find the square roots of large numbers by factoring them.
Consider an example. Find \(\sqrt(44100)\) . Since \(44100:100=441\) , then \(44100=100\cdot 441\) . According to the criterion of divisibility, the number \(441\) is divisible by \(9\) (since the sum of its digits is 9 and is divisible by 9), therefore, \(441:9=49\) , that is, \(441=9\ cdot 49\) .
Thus, we got: \[\sqrt(44100)=\sqrt(9\cdot 49\cdot 100)= \sqrt9\cdot \sqrt(49)\cdot \sqrt(100)=3\cdot 7\cdot 10=210\] Let's look at another example: \[\sqrt(\dfrac(32\cdot 294)(27))= \sqrt(\dfrac(16\cdot 2\cdot 3\cdot 49\cdot 2)(9\cdot 3))= \sqrt( \ dfrac(16\cdot4\cdot49)(9))=\dfrac(\sqrt(16)\cdot \sqrt4 \cdot \sqrt(49))(\sqrt9)=\dfrac(4\cdot 2\cdot 7)3 =\dfrac(56)3\]
\(\bullet\) Let's show how to enter numbers under the square root sign using the example of the expression \(5\sqrt2\) (short for the expression \(5\cdot \sqrt2\) ). Since \(5=\sqrt(25)\) , then \ Note also that, for example,
1) \(\sqrt2+3\sqrt2=4\sqrt2\) ,
2) \(5\sqrt3-\sqrt3=4\sqrt3\)
3) \(\sqrt a+\sqrt a=2\sqrt a\) .

Why is that? Let's explain with example 1). As you already understood, we cannot somehow convert the number \(\sqrt2\) . Imagine that \(\sqrt2\) is some number \(a\) . Accordingly, the expression \(\sqrt2+3\sqrt2\) is nothing but \(a+3a\) (one number \(a\) plus three more of the same numbers \(a\) ). And we know that this is equal to four such numbers \(a\) , that is, \(4\sqrt2\) .

Fact 4.
\(\bullet\) It is often said “cannot extract the root” when it is not possible to get rid of the sign \(\sqrt () \ \) of the root (radical) when finding the value of some number. For example, you can root the number \(16\) because \(16=4^2\) , so \(\sqrt(16)=4\) . But to extract the root from the number \(3\) , that is, to find \(\sqrt3\) , it is impossible, because there is no such number that squared will give \(3\) .
Such numbers (or expressions with such numbers) are irrational. For example, numbers \(\sqrt3, \ 1+\sqrt2, \ \sqrt(15)\) etc. are irrational.
Also irrational are the numbers \(\pi\) (the number “pi”, approximately equal to \(3,14\) ), \(e\) (this number is called the Euler number, approximately equal to \(2,7\) ) etc.
\(\bullet\) Please note that any number will be either rational or irrational. And together all rational and all irrational numbers form a set called set of real (real) numbers. This set is denoted by the letter \(\mathbb(R)\) .
This means that all the numbers that we currently know are called real numbers.

Fact 5.
\(\bullet\) Modulus of a real number \(a\) is a non-negative number \(|a|\) equal to the distance from the point \(a\) to \(0\) on the real line. For example, \(|3|\) and \(|-3|\) are equal to 3, since the distances from the points \(3\) and \(-3\) to \(0\) are the same and equal to \(3 \) .
\(\bullet\) If \(a\) is a non-negative number, then \(|a|=a\) .
Example: \(|5|=5\) ; \(\qquad |\sqrt2|=\sqrt2\) . \(\bullet\) If \(a\) is a negative number, then \(|a|=-a\) .
Example: \(|-5|=-(-5)=5\) ; \(\qquad |-\sqrt3|=-(-\sqrt3)=\sqrt3\).
They say that for negative numbers, the module “eats” the minus, and positive numbers, as well as the number \(0\) , the module leaves unchanged.
BUT this rule only applies to numbers. If you have an unknown \(x\) (or some other unknown) under the module sign, for example, \(|x|\) , about which we do not know whether it is positive, equal to zero or negative, then get rid of the module we can not. In this case, this expression remains so: \(|x|\) . \(\bullet\) The following formulas hold: \[(\large(\sqrt(a^2)=|a|))\] \[(\large((\sqrt(a))^2=a)), \text( provided ) a\geqslant 0\] The following mistake is often made: they say that \(\sqrt(a^2)\) and \((\sqrt a)^2\) are the same thing. This is true only when \(a\) is a positive number or zero. But if \(a\) is a negative number, then this is not true. It suffices to consider such an example. Let's take the number \(-1\) instead of \(a\). Then \(\sqrt((-1)^2)=\sqrt(1)=1\) , but the expression \((\sqrt (-1))^2\) does not exist at all (because it is impossible under the root sign put negative numbers in!).
Therefore, we draw your attention to the fact that \(\sqrt(a^2)\) is not equal to \((\sqrt a)^2\) ! Example: 1) \(\sqrt(\left(-\sqrt2\right)^2)=|-\sqrt2|=\sqrt2\), because \(-\sqrt2<0\) ;

\(\phantom(00000)\) 2) \((\sqrt(2))^2=2\) . \(\bullet\) Since \(\sqrt(a^2)=|a|\) , then \[\sqrt(a^(2n))=|a^n|\] (the expression \(2n\) denotes an even number)
That is, when extracting the root from a number that is in some degree, this degree is halved.
Example:
1) \(\sqrt(4^6)=|4^3|=4^3=64\)
2) \(\sqrt((-25)^2)=|-25|=25\) (note that if the module is not set, then it turns out that the root of the number is equal to \(-25\) ; but we remember , which, by definition of the root, this cannot be: when extracting the root, we should always get a positive number or zero)
3) \(\sqrt(x^(16))=|x^8|=x^8\) (since any number to an even power is non-negative)

Fact 6.
How to compare two square roots?
\(\bullet\) True for square roots: if \(\sqrt a<\sqrt b\) , то \(aExample:
1) compare \(\sqrt(50)\) and \(6\sqrt2\) . First, we transform the second expression into \(\sqrt(36)\cdot \sqrt2=\sqrt(36\cdot 2)=\sqrt(72)\). Thus, since \(50<72\) , то и \(\sqrt{50}<\sqrt{72}\) . Следовательно, \(\sqrt{50}<6\sqrt2\) .
2) Between which integers is \(\sqrt(50)\) ?
Since \(\sqrt(49)=7\) , \(\sqrt(64)=8\) , and \(49<50<64\) , то \(7<\sqrt{50}<8\) , то есть число \(\sqrt{50}\) находится между числами \(7\) и \(8\) .
3) Compare \(\sqrt 2-1\) and \(0,5\) . Suppose \(\sqrt2-1>0.5\) : \[\begin(aligned) &\sqrt 2-1>0.5 \ \big| +1\quad \text((add one to both sides))\\ &\sqrt2>0.5+1 \ \big| \ ^2 \quad\text((square both parts))\\ &2>1,5^2\\ &2>2,25 \end(aligned)\] We see that we have obtained an incorrect inequality. Therefore, our assumption was wrong and \(\sqrt 2-1<0,5\) .
Note that adding a certain number to both sides of the inequality does not affect its sign. Multiplying/dividing both sides of an inequality by a positive number also does not change its sign, but multiplying/dividing by a negative number reverses the sign of the inequality!
Both sides of an equation/inequality can be squared ONLY IF both sides are non-negative. For example, in the inequality from the previous example, you can square both sides, in the inequality \(-3<\sqrt2\) нельзя (убедитесь в этом сами)! \(\bullet\) Note that \[\begin(aligned) &\sqrt 2\approx 1,4\\ &\sqrt 3\approx 1,7 \end(aligned)\] Knowing the approximate meaning of these numbers will help you when comparing numbers! \(\bullet\) In order to extract the root (if it is extracted) from some large number that is not in the table of squares, you must first determine between which “hundreds” it is, then between which “tens”, and then determine the last digit of this number. Let's show how it works with an example.
Take \(\sqrt(28224)\) . We know that \(100^2=10\,000\) , \(200^2=40\,000\) and so on. Note that \(28224\) is between \(10\,000\) and \(40\,000\) . Therefore, \(\sqrt(28224)\) is between \(100\) and \(200\) .
Now let's determine between which “tens” our number is (that is, for example, between \(120\) and \(130\) ). We also know from the table of squares that \(11^2=121\) , \(12^2=144\) etc., then \(110^2=12100\) , \(120^2=14400 \) , \(130^2=16900\) , \(140^2=19600\) , \(150^2=22500\) , \(160^2=25600\) , \(170^2=28900 \) . So we see that \(28224\) is between \(160^2\) and \(170^2\) . Therefore, the number \(\sqrt(28224)\) is between \(160\) and \(170\) .
Let's try to determine the last digit. Let's remember what single-digit numbers when squaring give at the end \ (4 \) ? These are \(2^2\) and \(8^2\) . Therefore, \(\sqrt(28224)\) will end in either 2 or 8. Let's check this. Find \(162^2\) and \(168^2\) :
\(162^2=162\cdot 162=26224\)
\(168^2=168\cdot 168=28224\) .
Hence \(\sqrt(28224)=168\) . Voila!

In order to adequately solve the exam in mathematics, first of all, it is necessary to study the theoretical material, which introduces numerous theorems, formulas, algorithms, etc. At first glance, it may seem that this is quite simple. However, finding a source in which the theory for the Unified State Examination in mathematics is presented easily and understandably for students with any level of training is, in fact, a rather difficult task. School textbooks cannot always be kept at hand. And finding the basic formulas for the exam in mathematics can be difficult even on the Internet.

Why is it so important to study theory in mathematics, not only for those who take the exam?

1. Because it broadens your horizons. The study of theoretical material in mathematics is useful for anyone who wants to get answers to a wide range of questions related to the knowledge of the world. Everything in nature is ordered and has a clear logic. This is precisely what is reflected in science, through which it is possible to understand the world.
2. Because it develops the intellect. Studying reference materials for the exam in mathematics, as well as solving various problems, a person learns to think and reason logically, to formulate thoughts correctly and clearly. He develops the ability to analyze, generalize, draw conclusions.

We invite you to personally evaluate all the advantages of our approach to the systematization and presentation of educational materials.

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; four; 5; 6; 9.
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.

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When solving various problems from the course of mathematics and physics, pupils and students are often faced with the need to extract roots of the second, third or nth degree. Of course, in the century information technologies It will not be difficult to solve such a problem using a calculator. However, there are situations when it is impossible to use an electronic assistant.

For example, it is forbidden to bring electronics to many exams. In addition, the calculator may not be at hand. In such cases, it is useful to know at least some methods for manually calculating radicals.

One of the simplest ways to calculate roots is to using a special table. What is it and how to use it correctly?

Using the table, you can find the square of any number from 10 to 99. At the same time, the rows of the table contain tens values, and the columns contain unit values. The cell at the intersection of a row and a column contains a square two-digit number. In order to calculate the square of 63, you need to find a row with a value of 6 and a column with a value of 3. At the intersection, we find a cell with the number 3969.

Since extracting the root is the inverse operation of squaring, to perform this action, you must do the opposite: first find the cell with the number whose radical you want to calculate, then determine the answer from the column and row values. As an example, consider the calculation of the square root of 169.

We find a cell with this number in the table, horizontally we determine the tens - 1, vertically we find the units - 3. Answer: √169 = 13.

Similarly, you can calculate the roots of the cubic and n-th degree, using the appropriate tables.

The advantage of the method is its simplicity and the absence of additional calculations. The disadvantages are obvious: the method can only be used for a limited range of numbers (the number for which the root is found must be between 100 and 9801). In addition, it will not work if the given number is not in the table.

Prime factorization

If the table of squares is not at hand or with its help it was impossible to find the root, you can try decompose the number under the root into prime factors. Prime factors are those that can be completely (without remainder) divided only by itself or by one. Examples would be 2, 3, 5, 7, 11, 13, etc.

Consider the calculation of the root using the example √576. Let's decompose it into simple factors. We get the following result: √576 = √(2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 ​​∙ 3) = √(2 ∙ 2 ∙ 2)² ∙ √3². Using the main property of the roots √a² = a, we get rid of the roots and squares, after which we calculate the answer: 2 ∙ 2 ∙ 2 ∙ 3 ​​= 24.

What to do if any of the factors does not have its own pair? For example, consider the calculation of √54. After factoring, we get the result in the following form: The non-removable part can be left under the root. For most problems in geometry and algebra, such an answer will be counted as the final one. But if there is a need to calculate approximate values, you can use the methods that will be discussed later.

Heron's method

What to do when you need to know at least approximately what the extracted root is (if it is impossible to get an integer value)? A quick and fairly accurate result is obtained by applying the Heron method.. Its essence lies in the use of an approximate formula:

√R = √a + (R - a) / 2√a,

where R is the number whose root is to be calculated, a is the nearest number whose root value is known.

Let's see how the method works in practice and evaluate how accurate it is. Let's calculate what √111 is equal to. The nearest number to 111, the root of which is known, is 121. Thus, R = 111, a = 121. Substitute the values ​​in the formula:

√111 = √121 + (111 - 121) / 2 ∙ √121 = 11 - 10 / 22 ≈ 10,55.

Now let's check the accuracy of the method:

10.55² = 111.3025.

The error of the method was approximately 0.3. If the accuracy of the method needs to be improved, you can repeat the steps described earlier:

√111 = √111,3025 + (111 - 111,3025) / 2 ∙ √111,3025 = 10,55 - 0,3025 / 21,1 ≈ 10,536.

Let's check the accuracy of the calculation:

10.536² = 111.0073.

After repeated application of the formula, the error became quite insignificant.

Calculation of the root by division into a column

This method of finding the square root value is a little more complicated than the previous ones. However, it is the most accurate among other calculation methods without a calculator..

Let's say that you need to find the square root with an accuracy of 4 decimal places. Let's analyze the calculation algorithm using the example of an arbitrary number 1308.1912.

1. Divide the sheet of paper into 2 parts with a vertical line, and then draw another line from it to the right, slightly below the top edge. We write the number on the left side, dividing it into groups of 2 digits, moving to the right and left of the decimal point. The very first digit on the left can be without a pair. If the sign is missing on the right side of the number, then 0 should be added. In our case, we get 13 08.19 12.
2. Let's select the largest number whose square will be less than or equal to the first group of digits. In our case, this is 3. Let's write it on the top right; 3 is the first digit of the result. At the bottom right, we indicate 3 × 3 = 9; this will be needed for subsequent calculations. Subtract 9 from 13 in a column, we get the remainder 4.
3. Let's add the next pair of numbers to the remainder 4; we get 408.
4. Multiply the number on the top right by 2 and write it on the bottom right, adding _ x _ = to it. We get 6_ x _ =.
5. Instead of dashes, you need to substitute the same number, less than or equal to 408. We get 66 × 6 \u003d 396. Let's write 6 on the top right, since this is the second digit of the result. Subtract 396 from 408, we get 12.
6. Let's repeat steps 3-6. Since the digits carried down are in the fractional part of the number, it is necessary to put a decimal point on the top right after 6. Let's write the doubled result with dashes: 72_ x _ =. A suitable number would be 1: 721 × 1 = 721. Let's write it down as an answer. Let's subtract 1219 - 721 = 498.
7. Let's perform the sequence of actions given in the previous paragraph three more times to get the required number of decimal places. If there are not enough signs for further calculations, two zeros must be added to the current number on the left.

As a result, we get the answer: √1308.1912 ≈ 36.1689. If you check the action with a calculator, you can make sure that all the characters were determined correctly.

Bitwise calculation of the square root value

The method is highly accurate. In addition, it is quite understandable and it does not require memorizing formulas or a complex algorithm of actions, since the essence of the method is to select the correct result.

Let's extract the root from the number 781. Let's consider in detail the sequence of actions.

1. Find out which digit of the square root value will be the highest. To do this, let's square 0, 10, 100, 1000, etc. and find out between which of them the root number is located. We get that 10²< 781 < 100², т. е. старшим разрядом будут десятки.
2. Let's take the value of tens. To do this, we will take turns raising to the power of 10, 20, ..., 90, until we get a number greater than 781. For our case, we get 10² = 100, 20² = 400, 30² = 900. The value of the result n will be within 20< n <30.
3. Similarly to the previous step, the value of the units digit is selected. We alternately square 21.22, ..., 29: 21² = 441, 22² = 484, 23² = 529, 24² = 576, 25² = 625, 26² = 676, 27² = 729, 28² = 784. We get that 27< n < 28.
4. Each subsequent digit (tenths, hundredths, etc.) is calculated in the same way as shown above. Calculations are carried out until the required accuracy is achieved.

Extracting a root from a large number. Dear friends!In this article, we will show you how to take the root of a large number without a calculator. This is necessary not only for solving certain types of USE problems (there are those for movement), but also for the general mathematical development, it is desirable to know this analytical technique.

It would seem that everything is simple: factorize and extract. There is no problem. For example, the number 291600, when expanded, will give the product:

We calculate:

There is one BUT! The method is good if divisors 2, 3, 4 and so on are easily determined. But what if the number from which we extract the root is a product of prime numbers? For example, 152881 is the product of the numbers 17, 17, 23, 23. Try to find these divisors right away.

The essence of the method we are considering- this is pure analysis. The root with the accumulated skill is found quickly. If the skill is not worked out, but the approach is simply understood, then it is a little slower, but still determined.

Let's take the root of 190969.

First, let's determine between what numbers (multiples of a hundred) our result lies.

Obviously, the result of the root of a given number lies in the range from 400 to 500, because

400 2 =160000 and 500 2 =250000

Really:

in the middle, closer to 160,000 or 250,000?

The number 190969 is somewhere in the middle, but still closer to 160000. We can conclude that the result of our root will be less than 450. Let's check:

Indeed, it is less than 450, since 190,969< 202 500.

Now let's check the number 440:

So our result is less than 440, since 190 969 < 193 600.

Checking the number 430:

We have established that the result of this root lies in the range from 430 to 440.

The product of numbers ending in 1 or 9 gives a number ending in 1. For example, 21 times 21 equals 441.

The product of numbers ending in 2 or 8 gives a number ending in 4. For example, 18 times 18 equals 324.

The product of numbers ending in 5 gives a number ending in 5. For example, 25 times 25 equals 625.

The product of numbers ending in 4 or 6 gives a number ending in 6. For example, 26 times 26 equals 676.

The product of numbers ending in 3 or 7 gives a number ending in 9. For example, 17 times 17 equals 289.

Since the number 190969 ends with the number 9, then this product is either the number 433 or 437.

*Only they, when squared, can give 9 at the end.

We check:

So the result of the root will be 437.

That is, we kind of "felt" the right answer.

As you can see, the maximum that is required is to carry out 5 actions in a column. Perhaps you will immediately get to the point, or you will do just three actions. It all depends on how accurately you make the initial estimate of the number.

Extract your own root from 148996

Such a discriminant is obtained in the problem:

The motor ship passes along the river to the destination 336 km and after parking returns to the point of departure. Find the speed of the ship in still water, if the speed of the current is 5 km / h, the parking lasts 10 hours, and the ship returns to the point of departure 48 hours after leaving it. Give your answer in km/h.

View Solution

The result of the root is between the numbers 300 and 400:

300 2 =90000 400 2 =160000

Indeed, 90000<148996<160000.

The essence of further reasoning is to determine how the number 148996 is located (distanced) relative to these numbers.

Calculate the differences 148996 - 90000=58996 and 160000 - 148996=11004.

It turns out that 148996 is close (much closer) to 160000. Therefore, the result of the root will definitely be greater than 350 and even 360.

We can conclude that our result is greater than 370. Further, it is clear: since 148996 ends with the number 6, this means that you must square the number ending either in 4 or 6. *Only these numbers, when squared, give in end 6.

Sincerely, Alexander Krutitskikh.

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