The area of a square plot of land is 81 dm². Find his side. Suppose the side length of the square is X decimeters. Then the area of the plot is X² square decimeters. Since, according to the condition, this area is equal to 81 dm², then X² = 81. The length of a side of a square is a positive number. A positive number whose square is 81 is the number 9. When solving the problem, it was necessary to find the number x whose square is 81, i.e. solve the equation X² = 81. This equation has two roots: x 1 = 9 and x 2 = - 9, since 9² = 81 and (- 9)² = 81. Both numbers 9 and - 9 are called the square roots of 81.
Note that one of the square roots X= 9 is a positive number. It is called the arithmetic square root of 81 and is denoted √81, so √81 = 9.
Arithmetic square root of a number A is a non-negative number whose square is equal to A.
For example, the numbers 6 and - 6 are square roots of the number 36. However, the number 6 is an arithmetic square root of 36, since 6 is a non-negative number and 6² = 36. The number - 6 is not arithmetic root.
Arithmetic square root of a number A denoted as follows: √ A.
The sign is called the arithmetic sign square root; A- called a radical expression. Expression √ A read like this: arithmetic square root of a number A. For example, √36 = 6, √0 = 0, √0.49 = 0.7. In cases where it is clear that we are talking about an arithmetic root, they briefly say: “the square root of A«.
The act of finding the square root of a number is called square rooting. This action is the reverse of squaring.
You can square any number, but you can't extract square roots from any number. For example, it is impossible to extract the square root of the number - 4. If such a root existed, then, denoting it with the letter X, we would get the incorrect equality x² = - 4, since there is a non-negative number on the left and a negative number on the right.
Expression √ A only makes sense when a ≥ 0. The definition of square root can be briefly written as follows: √ a ≥ 0, (√A)² = A. Equality (√ A)² = A valid for a ≥ 0. Thus, to ensure that the square root of not negative number A equals b, i.e. in the fact that √ A =b, you need to check that the following two conditions are met: b ≥ 0, b² = A.
Square root of a fraction
Let's calculate. Note that √25 = 5, √36 = 6, and let’s check whether the equality holds.
Because 
and , then the equality is true. So, 
.
Theorem: If A≥ 0 and b> 0, that is, the root of the fraction is equal to the root of the numerator divided by the root of the denominator. It is required to prove that: and 
.
Since √ A≥0 and √ b> 0, then .
On the property of raising a fraction to a power and the definition of a square root 
the theorem is proven. Let's look at a few examples.
Calculate using the proven theorem 
.
Second example: Prove that 
, If A ≤ 0, b < 0. 
.
Another example: Calculate .
.
Square Root Conversion
Removing the multiplier from under the root sign. Let the expression be given. If A≥ 0 and b≥ 0, then using the product root theorem we can write:
This transformation is called removing the factor from the root sign. Let's look at an example;
Calculate at X= 2. Direct substitution X= 2 in the radical expression leads to complex calculations. These calculations can be simplified if you first remove the factors from under the root sign: . Substituting now x = 2, we get:.
So, when removing the factor from under the root sign, the radical expression is represented in the form of a product in which one or more factors are squares of non-negative numbers. Then apply the product root theorem and take the root of each factor. Let's consider an example: Simplify the expression A = √8 + √18 - 4√2 by taking out the factors in the first two terms from under the root sign, we get:. Let us emphasize that equality 
valid only when A≥ 0 and b≥ 0. if A < 0, то .
Exponentiation involves multiplying a given number by itself a certain number of times. For example, raising the number 2 to the fifth power would look like this:
The number that needs to be multiplied by itself is called the base of the power, and the number of multiplications is called its exponent. Raising to a power corresponds to two opposite actions: finding the exponent and finding the base.
Root extraction
Finding the base of a power is called root extraction. This means that you need to find the number that needs to be raised to the power n to get the given one.
For example, it is necessary to extract the 4th root of the number 16, i.e. to determine, you need to multiply by itself 4 times to ultimately get 16. This number is 2.
This arithmetic operation is written using special sign– radical: √, above which the exponent is indicated on the left.
Arithmetic root
If the exponent is an even number, then the root can be two numbers with the same absolute value, but c is positive and negative. So, in the example given, these could be the numbers 2 and -2.
The expression must be unambiguous, i.e. have one result. For this purpose, the concept of an arithmetic root was introduced, which can only represent a positive number. An arithmetic root cannot be less than zero.
Thus, in the example discussed above, only the number 2 will be the arithmetic root, and the second answer option - -2 - is excluded by definition.
Square root
For some degrees, which are used more often than others, there are special names that are originally associated with geometry. We are talking about raising to the second and third powers.
To the second power the length of a side of a square when you need to calculate its area. If you need to find the volume of a cube, the length of its edge is raised to the third power. Therefore it is called the square of the number, and the third is called the cube.
Accordingly, the root of the second degree is called square, and the root of the third degree is called cubic. The square root is the only root that is not written with an exponent above the radical:
So, the arithmetic square root of given number is a positive number that must be raised to the second power to obtain the given number.
It's time to sort it out root extraction methods. They are based on the properties of roots, in particular, on the equality, which is true for any non-negative number b.
Below we will look at the main methods of extracting roots one by one.
Let's start with the simplest case - extracting roots from natural numbers using a table of squares, a table of cubes, etc.
If tables of squares, cubes, etc. If you don’t have it at hand, it’s logical to use the method of extracting the root, which involves decomposing the radical number into prime factors.
It is worth special mentioning what is possible for roots with odd exponents.
Finally, let's consider a method that allows us to sequentially find the digits of the root value.
Let's get started.
Using a table of squares, a table of cubes, etc.
In the simplest cases, tables of squares, cubes, etc. allow you to extract roots. What are these tables?
The table of squares of integers from 0 to 99 inclusive (shown below) consists of two zones. The first zone of the table is located on a gray background; by selecting a specific row and a specific column, it allows you to compose a number from 0 to 99. For example, let’s select a row of 8 tens and a column of 3 units, with this we fixed the number 83. The second zone occupies the rest of the table. Each cell is located at the intersection of a certain row and a certain column, and contains the square of the corresponding number from 0 to 99. At the intersection of our chosen row of 8 tens and column 3 of ones there is a cell with the number 6,889, which is the square of the number 83.
Tables of cubes, tables of fourth powers of numbers from 0 to 99, and so on are similar to the table of squares, only they contain cubes, fourth powers, etc. in the second zone. corresponding numbers.
Tables of squares, cubes, fourth powers, etc. allow you to extract square roots, cube roots, fourth roots, etc. accordingly from the numbers in these tables. Let us explain the principle of their use when extracting roots.
Let's say we need to extract the nth root of the number a, while the number a is contained in the table of nth powers. Using this table we find the number b such that a=b n. Then 
, therefore, the number b will be the desired root of the nth degree.
As an example, let's show how to use a cube table to extract the cube root of 19,683. We find the number 19,683 in the table of cubes, from it we find that this number is the cube of the number 27, therefore, 
.
It is clear that tables of nth powers are very convenient for extracting roots. However, they are often not at hand, and compiling them requires some time. Moreover, it is often necessary to extract roots from numbers that are not contained in the corresponding tables. In these cases, you have to resort to other methods of root extraction.
Factoring a radical number into prime factors
Enough in a convenient way, which makes it possible to extract a root from a natural number (if, of course, the root is extracted), is the decomposition of the radical number into prime factors. His the point is this: after that it is quite easy to represent it as a power with the desired exponent, which allows you to obtain the value of the root. Let's clarify this point.
Let the nth root of a natural number a be taken and its value equal b. In this case, the equality a=b n is true. Number b like any natural number can be represented as the product of all its prime factors p 1 , p 2 , …, p m in the form p 1 · p 2 · … · p m , and the radical number a in this case is represented as (p 1 · p 2 · … · p m) n. Since the decomposition of a number into prime factors is unique, the decomposition of the radical number a into prime factors will have the form (p 1 ·p 2 ·…·p m) n, which makes it possible to calculate the value of the root as.
Note that if the decomposition into prime factors of a radical number a cannot be represented in the form (p 1 ·p 2 ·…·p m) n, then the nth root of such a number a is not completely extracted.
Let's figure this out when solving examples.
Example.
Take the square root of 144.
Solution.
If you look at the table of squares given in the previous paragraph, you can clearly see that 144 = 12 2, from which it is clear that the square root of 144 is equal to 12.
But in light of this point, we are interested in how the root is extracted by decomposing the radical number 144 into prime factors. Let's look at this solution.
Let's decompose 144 to prime factors:
That is, 144=2·2·2·2·3·3. Based on the resulting decomposition, the following transformations can be carried out: 144=2·2·2·2·3·3=(2·2) 2·3 2 =(2·2·3) 2 =12 2. Hence, 
.
Using the properties of the degree and the properties of the roots, the solution could be formulated a little differently: .
Answer:
To consolidate the material, consider the solutions to two more examples.
Example.
Calculate the value of the root.
Solution.
The prime factorization of the radical number 243 has the form 243=3 5 . Thus, 
.
Answer:
Example.
Is the root value an integer?
Solution.
To answer this question, let's factor the radical number into prime factors and see if it can be represented as a cube of an integer.
We have 285 768=2 3 ·3 6 ·7 2. The resulting expansion is not represented as a cube of an integer, since the degree prime factor 7 is not a multiple of three. Therefore, the cube root of 285,768 cannot be extracted completely.
Answer:
No.
Extracting roots from fractional numbers
It's time to figure out how to extract the root from fractional number. Let the fractional radical number be written as p/q. According to the property of the root of a quotient, the following equality is true. From this equality it follows rule for extracting the root of a fraction: The root of a fraction is equal to the quotient of the root of the numerator divided by the root of the denominator.
Let's look at an example of extracting a root from a fraction.
Example.
What is the square root of common fraction 25/169 .
Solution.
Using the table of squares, we find that the square root of the numerator of the original fraction is equal to 5, and the square root of the denominator is equal to 13. Then 
. This completes the extraction of the root of the common fraction 25/169.
Answer:
The root of a decimal fraction or mixed number is extracted after replacing the radical numbers with ordinary fractions.
Example.
Take the cube root of the decimal fraction 474.552.
Solution.
Let's imagine the original decimal as a common fraction: 474.552=474552/1000. Then 
. It remains to extract the cube roots that are in the numerator and denominator of the resulting fraction. Because 474 552=2 2 2 3 3 3 13 13 13=(2 3 13) 3 =78 3 and 1 000 = 10 3, then 
And 
. All that remains is to complete the calculations 
.
Answer:
.
Taking the root of a negative number
It is worthwhile to dwell on extracting roots from negative numbers. When studying roots, we said that when the root exponent is an odd number, then there can be a negative number under the root sign. We gave these entries the following meaning: for a negative number −a and an odd exponent of the root 2 n−1, 
. This equality gives rule for extracting odd roots from negative numbers: to extract the root of a negative number, you need to take the root of the opposite positive number, and put a minus sign in front of the result.
Let's look at the example solution.
Example.
Find the value of the root.
Solution.
Let's transform the original expression so that there is a positive number under the root sign: 
. Now mixed number replace it with an ordinary fraction: 
. We apply the rule for extracting the root of an ordinary fraction: 
. It remains to calculate the roots in the numerator and denominator of the resulting fraction: 
.
Here is a short summary of the solution: 
.
Answer:
.
Bitwise determination of the root value
In the general case, under the root there is a number that, using the techniques discussed above, cannot be represented as the nth power of any number. But in this case there is a need to know the meaning of a given root, at least up to a certain sign. In this case, to extract the root, you can use an algorithm that allows you to consistently obtain sufficient quantity values of the digits of the required number.
On the first step of this algorithm you need to find out what the most significant bit of the root value is. To do this, the numbers 0, 10, 100, ... are sequentially raised to the power n until the moment when a number exceeds the radical number is obtained. Then the number that we raised to the power n at the previous stage will indicate the corresponding most significant digit.
For example, consider this step of the algorithm when extracting the square root of five. We take the numbers 0, 10, 100, ... and square them until we get a number greater than 5. We have 0 2 =0<5 , 10 2 =100>5, which means the most significant digit will be the ones digit. The value of this bit, as well as the lower ones, will be found in the next steps of the root extraction algorithm.
All subsequent steps of the algorithm are aimed at sequentially clarifying the value of the root by finding the values of the next bits of the desired value of the root, starting with the highest one and moving to the lowest ones. For example, the value of the root at the first step turns out to be 2, at the second – 2.2, at the third – 2.23, and so on 2.236067977…. Let us describe how the values of the digits are found.
The digits are found by searching through them possible values 0, 1, 2, …, 9. In this case, the nth powers of the corresponding numbers are calculated in parallel, and they are compared with the radical number. If at some stage the value of the degree exceeds the radical number, then the value of the digit corresponding to the previous value is considered found, and the transition to the next step of the root extraction algorithm is made; if this does not happen, then the value of this digit is 9.
Let us explain these points using the same example of extracting the square root of five.
First we find the value of the units digit. We will go through the values 0, 1, 2, ..., 9, calculating 0 2, 1 2, ..., 9 2, respectively, until we get a value greater than the radical number 5. It is convenient to present all these calculations in the form of a table: 
So the value of the units digit is 2 (since 2 2<5
, а 2 3 >5 ). Let's move on to finding the value of the tenths place. In this case, we will square the numbers 2.0, 2.1, 2.2, ..., 2.9, comparing the resulting values with the radical number 5: 
Since 2.2 2<5
, а 2,3 2 >5, then the value of the tenths place is 2. You can proceed to finding the value of the hundredths place: 
This is how the next value of the root of five was found, it is equal to 2.23. And so you can continue to find values: 2,236, 2,2360, 2,23606, 2,236067, … .
To consolidate the material, we will analyze the extraction of the root with an accuracy of hundredths using the considered algorithm.
First we determine the most significant digit. To do this, we cube the numbers 0, 10, 100, etc. until we get a number greater than 2,151,186. We have 0 3 =0<2 151,186 , 10 3 =1 000<2151,186 , 100 3 =1 000 000>2 151,186 , so the most significant digit is the tens digit.
Let's determine its value. 
Since 10 3<2 151,186
, а 20 3 >2 151.186, then the value of the tens place is 1. Let's move on to units. 
Thus, the value of the ones digit is 2. Let's move on to tenths. 
Since even 12.9 3 is less than the radical number 2 151.186, then the value of the tenths place is 9. It remains to perform the last step of the algorithm; it will give us the value of the root with the required accuracy. 
At this stage, the value of the root is found accurate to hundredths: 
.
In conclusion of this article, I would like to say that there are many other ways to extract roots. But for most tasks, the ones we studied above are sufficient.
Bibliography.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8th grade. educational institutions.
- Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the beginnings of analysis: Textbook for grades 10 - 11 of general education institutions.
- Gusev V.A., Mordkovich A.G. Mathematics (a manual for those entering technical schools).
Fact 1.
\(\bullet\) Let's take some non-negative number \(a\) (that is, \(a\geqslant 0\) ). Then (arithmetic) square root from the number \(a\) is called such a non-negative number \(b\) , when squared we get the number \(a\) : \[\sqrt a=b\quad \text(same as )\quad a=b^2\] From the definition it follows 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 \(\sqrt(25)\) equal to? We know that \(5^2=25\) and \((-5)^2=25\) . Since by definition we must find a non-negative number, then \(-5\) is not suitable, therefore, \(\sqrt(25)=5\) (since \(25=5^2\) ).
Finding the value of \(\sqrt a\) is called taking the square root of the number \(a\) , and the number \(a\) is called the radical expression.
\(\bullet\) Based on the definition, expression \(\sqrt(-25)\), \(\sqrt(-4)\), 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 operations can you do with square roots?
\(\bullet\) The sum or difference of square roots IS NOT EQUAL to the square root of the sum or difference, that is \[\sqrt a\pm\sqrt b\ne \sqrt(a\pm b)\] Thus, if you need to calculate, for example, \(\sqrt(25)+\sqrt(49)\) , then initially you must find the values of \(\sqrt(25)\) and \(\sqrt(49)\ ) and then fold them. Hence, \[\sqrt(25)+\sqrt(49)=5+7=12\] If the values \(\sqrt a\) or \(\sqrt b\) cannot be found when adding \(\sqrt a+\sqrt b\), then such an expression is not transformed further and remains as it is. For example, in the sum \(\sqrt 2+ \sqrt (49)\) we can find \(\sqrt(49)\) is \(7\) , but \(\sqrt 2\) cannot be transformed in any way, That's why \(\sqrt 2+\sqrt(49)=\sqrt 2+7\). Unfortunately, this expression cannot be simplified further\(\bullet\) The product/quotient of square roots is equal to the square root of the product/quotient, that is \[\sqrt a\cdot \sqrt b=\sqrt(ab)\quad \text(s)\quad \sqrt a:\sqrt b=\sqrt(a:b)\] (provided that both sides 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 square roots of large numbers by factoring them.
Let's look at an example. Let's 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 notation for the expression \(5\cdot \sqrt2\)). Since \(5=\sqrt(25)\) , then
Note also that, for example,
1) \(\sqrt2+3\sqrt2=4\sqrt2\) ,
2) \(5\sqrt3-\sqrt3=4\sqrt3\)
Why is that? Let's explain using example 1). As you already understand, we cannot somehow transform the number \(\sqrt2\). Let's imagine that \(\sqrt2\) is some number \(a\) . Accordingly, the expression \(\sqrt2+3\sqrt2\) is nothing more than \(a+3a\) (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.
\(\bullet\) They often say “you can’t extract the root” when you can’t get rid of the sign \(\sqrt () \ \) of the root (radical) when finding the value of a number. For example, you can take the root of the number \(16\) because \(16=4^2\) , therefore \(\sqrt(16)=4\) . But it is impossible to extract the root of the number \(3\), that is, to find \(\sqrt3\), because there is no number that squared will give \(3\) .
Such numbers (or expressions with such numbers) are irrational. For example, numbers \(\sqrt3, \ 1+\sqrt2, \ \sqrt(15)\) and so on. are irrational.
Also irrational are the numbers \(\pi\) (the number “pi”, approximately equal to \(3.14\)), \(e\) (this number is called the Euler number, it is 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 a set of 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.
\(\bullet\) The 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 modulus “eats” the minus, while positive numbers, as well as the number \(0\), are left unchanged by the modulus. This rule only applies to numbers. If under your modulus sign there is an unknown \(x\) (or some other unknown), for example, \(|x|\) , about which we do not know whether it is positive, zero or negative, then get rid of the modulus we can not. In this case, this expression remains the same: \(|x|\) . \(\bullet\) The following formulas hold: \[(\large(\sqrt(a^2)=|a|))\]\[(\large((\sqrt(a))^2=a)), \text( provided ) a\geqslant 0\]
Very often the following mistake is made: they say that \(\sqrt(a^2)\) and \((\sqrt a)^2\) are the same thing. This is only true if \(a\) is a positive number or zero. But if \(a\) is a negative number, then this is false. It is enough to consider this example. Let's take instead of \(a\) the number \(-1\) . Then \(\sqrt((-1)^2)=\sqrt(1)=1\) , but the expression \((\sqrt (-1))^2\) does not exist at all (after all, it is impossible to use the root sign put negative numbers!). Therefore, we draw your attention to the fact that \(\sqrt(a^2)\) is not equal to \((\sqrt a)^2\) ! Example: 1)\(\sqrt(\left(-\sqrt2\right)^2)=|-\sqrt2|=\sqrt2\)<0\)
;
, because \(-\sqrt2 \(\phantom(00000)\) 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 taking the root of a number that is to 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 supplied, it turns out that the root of the number is equal to \(-25\) ; but we remember , that by definition of a root this cannot happen: when extracting a 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.<\sqrt b\)
, то \(a
How to compare two square roots? \(\bullet\) For square roots it is true: if \(\sqrt a 1) compare \(\sqrt(50)\) and \(6\sqrt2\) . First, let's transform the second expression into<72\)
, то и \(\sqrt{50}<\sqrt{72}\)
. Следовательно, \(\sqrt{50}<6\sqrt2\)
.
\(\sqrt(36)\cdot \sqrt2=\sqrt(36\cdot 2)=\sqrt(72)\)
. Thus, since \(50<50<64\)
, то \(7<\sqrt{50}<8\)
, то есть число \(\sqrt{50}\)
находится между числами \(7\)
и \(8\)
.
2) Between what integers is \(\sqrt(50)\) located? Since \(\sqrt(49)=7\) , \(\sqrt(64)=8\) , and \(49 3) Let's compare \(\sqrt 2-1\) and \(0.5\) . Let's assume that \(\sqrt2-1>0.5\) :<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 affect its sign, but multiplying/dividing by a negative number reverses the sign of the inequality!
You can square both sides of an equation/inequality 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)!
\(\bullet\) It should be remembered 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 can be extracted) from some large number that is not in the table of squares, you must first determine between which “hundreds” it is located, then – between which “tens”, and then determine the last digit of this number. Let's show how this works with an example.
Let's take \(\sqrt(28224)\) . We know that \(100^2=10\,000\), \(200^2=40\,000\), etc. 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 located (that is, for example, between \(120\) and \(130\)). Also from the table of squares we know 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 squared, give \(4\) at the end? These are \(2^2\) and \(8^2\) . Therefore, \(\sqrt(28224)\) will end in either 2 or 8. Let's check this. Let's find \(162^2\) and \(168^2\) :
\(162^2=162\cdot 162=26224\)
\(168^2=168\cdot 168=28224\) .
In order to adequately solve the Unified State Exam in mathematics, you first need to study theoretical material, which introduces you to 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 Exam in mathematics is presented in an easy and understandable way for students with any level of training is in fact a rather difficult task. School textbooks cannot always be kept at hand. And finding basic formulas for the Unified State Exam in mathematics can be difficult even on the Internet.
Why is it so important to study theory in mathematics not only for those taking the Unified State Exam?
- Because it broadens your horizons. Studying theoretical material in mathematics is useful for anyone who wants to get answers to a wide range of questions related to knowledge of the world around them. 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.
- Because it develops intelligence. By studying reference materials for the Unified State Exam in mathematics, as well as solving various problems, a person learns to think and reason logically, to formulate thoughts competently and clearly. He develops the ability to analyze, generalize, and draw conclusions.
We invite you to personally evaluate all the advantages of our approach to systematization and presentation of educational materials.