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 square root and they 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 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?
\(\bullet\) Sum or difference square roots 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 the 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) \(5\sqrt3-\sqrt3=4\sqrt3\)

3) \(\sqrt a+\sqrt a=2\sqrt a\) .

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, \(4\sqrt2\) .
\(\bullet\) 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 everyone is rational and everything 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 are on this moment we know are called real numbers.

Fact 5.
\(\bullet\) The modulus of a real number \(a\) is a non-negative number \(|a|\) , equal to 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. BUT 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: \[(\large(\sqrt(a^2)=|a|))\] \[(\large((\sqrt(a))^2=a)), \text( provided ) a\geqslant 0\] 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) \(\sqrt(\left(-\sqrt2\right)^2)=|-\sqrt2|=\sqrt2\), 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 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.
How to compare two square roots?<\sqrt b\) , то \(aThat is, when taking the root of a number that is to some degree, this degree is halved.
\(\bullet\) For square roots it is true: if \(\sqrt a 1) compare \(\sqrt(50)\) and \(6\sqrt2\) . First, let's transform the second expression into\(\sqrt(36)\cdot \sqrt2=\sqrt(36\cdot 2)=\sqrt(72)\)<72\) , то и \(\sqrt{50}<\sqrt{72}\) . Следовательно, \(\sqrt{50}<6\sqrt2\) .
. Thus, since \(50
2) Between what integers is \(\sqrt(50)\) located?<50<64\) , то \(7<\sqrt{50}<8\) , то есть число \(\sqrt{50}\) находится между числами \(7\) и \(8\) .
Since \(\sqrt(49)=7\) , \(\sqrt(64)=8\) , and \(49 3) Let's compare \(\sqrt 2-1\) and \(0.5\) . Let's assume that \(\sqrt2-1>0.5\) :\[\begin(aligned) &\sqrt 2-1>0.5 \ \big| +1\quad \text((add one to both sides))\\ &\sqrt2>0.5+1 \ \big| \ ^2 \quad\text((squaring both sides))\\ &2>1.5^2\\ &2>2.25 \end(aligned)\]<0,5\) .
We see that we have obtained an incorrect inequality. Therefore, our assumption was incorrect and \(\sqrt 2-1
Note that adding a certain number to both sides of the inequality does not affect its sign. Multiplying/dividing both sides of an inequality by a positive number also does not affect its sign, but multiplying/dividing by a negative number reverses the sign of the inequality!<\sqrt2\) нельзя (убедитесь в этом сами)! 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 \(\bullet\) It should be remembered that 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 \(168^2\) :
\(162^2=162\cdot 162=26224\)
\(168^2=168\cdot 168=28224\) .

Therefore, \(\sqrt(28224)=168\) . Voila!

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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

A fairly convenient way to extract the root of a natural number (if, of course, the root is extracted) is to decompose 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 us 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. The 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 cannot be represented as a cube of an integer, since the power of the 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 of a 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 the 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 fraction as an ordinary 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 replace the mixed number 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 sequentially obtain a sufficient number of digit values ​​of the desired number.

The first step of this algorithm is 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. 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 their 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).

The nth root of a number is a number that, when raised to that power, gives the number from which the root is extracted. Most often, actions are performed with square roots, which correspond to 2 degrees. When extracting a root, it is often impossible to detect it clearly, and the result is a number that cannot be represented as a natural fraction (transcendental). But using some techniques, you can greatly simplify the solution of examples with roots.

You will need

• – representation of the root of a number;
• – actions with degrees;
• – abbreviated multiplication formulas;
• - calculator.

Instructions

1. If absolute accuracy is not required, use a calculator when solving examples with roots. In order to extract the square root of a number, type it on the keyboard and simply press the corresponding button, which shows the root sign. As usual, calculators take the square root. But to calculate the roots of higher powers, use the function of raising a number to a power (on an engineering calculator).

2. To find the square root, raise the number to the power of 1/2, the cube root to 1/3, and so on. At the same time, strictly consider that when extracting roots of even degrees, the number must be positive; on the contrary, the calculator will simply not produce a result. Is this due to the fact that when raised to an even power, every number will be positive, say, (-2)^4=(-2)? (-2)? (-2)? (-2)=16. To extract the entire square root, when possible, use the table of squares of natural numbers.

3. If you don’t have a calculator nearby, or you need unconditional accuracy in calculations, use the properties of roots, as well as different formulas to simplify expressions. It is possible to extract partial roots from many numbers. To do this, use the property that the root of the product of 2 numbers is equal to the product of the roots of these numbers?m?n=?m??n.

4. Example. Calculate the value of the expression (?80-?45)/?5. Direct calculation will not yield anything, since not a single root is extracted completely. Transform the expression (?16?5-?9?5)/ ?5=(?16??5-?9??5)/ ?5=?5?(?16-?9)/ ?5. Reduce the numerator and denominator by?5, you get (?16-?9)=4-3=1.

5. If the radical expression or the root itself is built into a degree, then when extracting the root, use the property that the exponent of the radical expression can be divided by the degree of the root. If division is performed entirely, the number is entered from under the root. Let's say ?5^4=5?=25. Example. Calculate the value of the expression (?3+?5)?(?3-?5). Apply the square difference formula and get (?3)?-(?5)?=3-5=-2.

An ordinary fraction is a capricious number. Occasionally one has to suffer in order to discover the solution to a problem with fraction and present it in the proper form. Having learned to decide examples With fraction, you can easily cope with this unpleasant thing.

Instructions

1. Review adding and subtracting fractions. For example, 5/2+10/5. Reduce both fractions to a common denominator. To do this, find the number that can be divided without a remainder by the denominator of both the first and second fractions. In our case, this is the number 10. Transform the above fractions, it turns out 25/10+20/10. Now add the numerators together, and leave the denominator unchanged. It turns out 45/10. You can reduce the resulting fraction, that is, divide the numerator and denominator by the same number. It turns out 9/2. Select the whole part. Find the highest number that can be divided without a remainder by the denominator. This number is 8. Divide it by the denominator - this will be the whole part. It turns out that the total is 4 1/2. Do the same thing when subtracting fractions.

2. Review multiplication of fractions. Everything here is primitive. Multiply the numerators and denominators together. For example, 2/5 multiplied by 4/2 equals 8/10. Reduce the fraction to get 4/5.

3. Look at dividing fractions. When performing this action, reverse one of the fractions, and then multiply the numerators and denominators. Let's say, 2/5 divided by 4/2 - you get 2/5 multiplied by 2/4 - you get 4/20. Reduce the fraction to get 1/5.

Video on the topic

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When solving some mathematical problems, you have to operate with square roots. Therefore, it is important to know the rules of operations with square roots and learn how to transform expressions containing them. The goal is to study the rules of operations with square roots and ways to transform expressions with square roots.

We know that some rational numbers are expressed as infinite periodic decimal fractions, such as the number 1/1998=0.000500500500... But nothing prevents us from imagining a number whose decimal expansion does not reveal any period. Such numbers are called irrational.

The history of irrational numbers dates back to the amazing discovery of the Pythagoreans back in the 6th century. BC e. It all started with a seemingly simple question: what number expresses the length of the diagonal of a square with side 1?

The diagonal divides the square into 2 identical right-angled triangles, in each of which it acts as a hypotenuse. Therefore, as follows from the Pythagorean theorem, the length of the diagonal of a square is equal to

. The temptation immediately arises to take out a microcalculator and press the square root key. On the scoreboard we will see 1.4142135. A more advanced calculator that performs calculations with high accuracy will show 1.414213562373. And with the help of a modern powerful computer you can calculate with an accuracy of hundreds, thousands, millions of decimal places. But even the most powerful computer, no matter how long it runs, will never be able to calculate all the decimal digits or detect any period in them.

And although Pythagoras and his students did not have a computer, they were the ones who substantiated this fact. The Pythagoreans proved that the diagonal of a square and its side have no common measure (i.e., a segment that would be plotted an integer number of times both on the diagonal and on the side). Therefore, the ratio of their lengths is the number

– cannot be expressed as the ratio of some integers m and n. And since this is so, we add, the decimal expansion of a number does not reveal any regular pattern.

Following the discovery of the Pythagoreans

How to prove that a number

irrational? Suppose there is a rational number m/n=. We will consider the fraction m/n irreducible, because a reducible fraction can always be reduced to an irreducible one. Raising both sides of the equality, we get . From here we conclude that m is an even number, that is, m = 2K. Therefore and, therefore, , or . But then we get that n is an even number, but this cannot be, since the fraction m/n is irreducible. A contradiction arises.

It remains to conclude that our assumption is incorrect and the rational number m/n is equal to

does not exist.

1. Square root of a number

Knowing the time t , you can find the path in free fall using the formula:

Let's solve the inverse problem.

Task . How many seconds will it take for a stone dropped from a height of 122.5 m to fall?

To find the answer, you need to solve the equation

From it we find that Now it remains to find a positive number t such that its square is 25. This number is 5, since So the stone will fall for 5 s.

You also have to look for a positive number by its square when solving other problems, for example, when finding the length of the side of a square by its area. Let us introduce the following definition.

Definition . A non-negative number whose square is equal to a non-negative number a is called the square root of a. This number stands for

Thus

Example . Because

You cannot take square roots from negative numbers, since the square of any number is either positive or equal to zero. For example, the expression

has no numerical value. the sign is called the radical sign (from the Latin “radix” - root), and the number A - radical number. For example, in the notation the radical number is 25. Since This means that the square root of the number written by one and 2n zeros, is equal to the number written by one and n

zeros: = 10…0

2n zeros n zeros

Similarly, it is proved that

2n zeros n zeros

For example,

2. Calculating square roots

We know that there is no rational number whose square is 2. This means that cannot be a rational number. It is an irrational number, i.e. is written as a non-periodic infinite decimal fraction, and the first decimal places of this fraction are 1.414... To find the next decimal place, you need to take the number 1.414 X cannot be a rational number. It is an irrational number, i.e. is written as a non-periodic infinite decimal fraction, and the first decimal places of this fraction are 1.414... To find the next decimal place, you need to take the number 1.414, Where can take the values ​​0, 1, 2, 3, 4, 5, 6, 7, 8, 9, square these numbers in order and find such a value X, in which the square is less than 2, but the next square is greater than 2. This value is x=2. cannot be a rational number. It is an irrational number, i.e. is written as a non-periodic infinite decimal fraction, and the first decimal places of this fraction are 1.414... To find the next decimal place, you need to take the number 1.414 Next, we repeat the same thing with numbers like 1.4142

. Continuing this process, we obtain one after another the digits of the infinite decimal fraction equal to .

with eight correct numbers. To do this, just enter the number into the microcalculator a>0 and press the key - 8 digits of the value will be displayed on the screen. In some cases it is necessary to use the properties of square roots, which we will indicate below.

If the accuracy provided by a microcalculator is insufficient, you can use the method for refining the value of the root given by the following theorem.

Theorem. If a is a positive number and is an approximate value for by excess, then