How to find the square root of a number manually. The root of a word - what it is and how to find it

Root formulas. Properties of square roots.

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In the previous lesson we figured out what a square root is. It's time to figure out which ones exist formulas for roots what are properties of roots, and what can be done with all this.

Formulas of roots, properties of roots and rules for working with roots- this is essentially the same thing. There are surprisingly few formulas for square roots. Which certainly makes me happy! Or rather, you can write a lot of different formulas, but for practical and confident work with roots, only three are enough. Everything else flows from these three. Although many people get confused in the three root formulas, yes...

Let's start with the simplest one. Here she is:

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You can get acquainted with functions and derivatives.

Phylogenetically, the root arose later than the stem and leaf - in connection with the transition of plants to life on land and probably originated from root-like underground branches. The root has neither leaves nor buds arranged in a certain order. It is characterized by apical growth in length, its lateral branches arise from internal tissues, the growth point is covered with a root cap. The root system is formed throughout the life of the plant organism. Sometimes the root can serve as a storage site for nutrients. In this case, it changes.

Types of roots

The main root is formed from the embryonic root during seed germination. Lateral roots extend from it.

Adventitious roots develop on stems and leaves.

Lateral roots are branches of any roots.

Each root (main, lateral, adventitious) has the ability to branch, which significantly increases the surface of the root system, and this helps to better strengthen the plant in the soil and improve its nutrition.

Types of root systems

There are two main types of root systems: taproot, which has a well-developed main root, and fibrous. The fibrous root system consists of a large number of adventitious roots, equal in size. The entire mass of roots consists of lateral or adventitious roots and has the appearance of a lobe.

The highly branched root system forms a huge absorbing surface. For example,

• the total length of winter rye roots reaches 600 km;
• length of root hairs - 10,000 km;
• the total root surface is 200 m2.

This is many times the area of ​​the aboveground mass.

If the plant has a well-defined main root and adventitious roots develop, then a mixed type root system (cabbage, tomato) is formed.

External structure of the root. Internal structure of the root

Root zones

Root cap

The root grows in length from its apex, where the young cells of the educational tissue are located. The growing part is covered with a root cap, which protects the root tip from damage and facilitates the movement of the root in the soil during growth. The latter function is carried out due to the property of the outer walls of the root cap being covered with mucus, which reduces friction between the root and soil particles. They can even push soil particles apart. The cells of the root cap are living and often contain starch grains. The cells of the cap are constantly renewed due to division. Participates in positive geotropic reactions (direction of root growth towards the center of the Earth).

Cells of the division zone are actively dividing, the length of this zone is different types and different roots of the same plant are not the same.

Behind the division zone is an extension zone (growth zone). The length of this zone does not exceed a few millimeters.

As linear growth completes, the third stage of root formation begins—its differentiation; a zone of cell differentiation and specialization (or a zone of root hairs and absorption) is formed. In this zone, the outer layer of the epiblema (rhizoderm) with root hairs, the layer of the primary cortex and the central cylinder are already distinguished.

Root hair structure

Root hairs are highly elongated outgrowths of the outer cells covering the root. The number of root hairs is very large (per 1 mm2 from 200 to 300 hairs). Their length reaches 10 mm. Hairs form very quickly (in young apple tree seedlings in 30-40 hours). Root hairs are short-lived. They die off after 10-20 days, and new ones grow on the young part of the root. This ensures the development of new soil horizons by the roots. The root continuously grows, forming more and more new areas of root hairs. Hairs can not only absorb ready-made solutions of substances, but also contribute to the dissolution of certain soil substances and then absorb them. The area of ​​the root where the root hairs have died is able to absorb water for a while, but then becomes covered with a plug and loses this ability.

The hair shell is very thin, which facilitates the absorption of nutrients. Almost the entire hair cell is occupied by a vacuole, surrounded by a thin layer of cytoplasm. The nucleus is at the top of the cell. A mucous sheath is formed around the cell, which promotes the gluing of root hairs to soil particles, which improves their contact and increases the hydrophilicity of the system. Absorption is facilitated by the secretion of acids (carbonic, malic, citric) by root hairs, which dissolve mineral salts.

Root hairs also play a mechanical role - they serve as support for the root tip, which passes between the soil particles.

Under a microscope, a cross section of the root in the absorption zone shows its structure at the cellular and tissue levels. On the surface of the root there is rhizoderm, under it there is bark. The outer layer of the cortex is the exodermis, inward from it is the main parenchyma. Its thin-walled living cells perform a storage function, conducting nutrient solutions in a radial direction - from the suction tissue to the vessels of the wood. They also contain the synthesis of a number of organic substances vital for the plant. The inner layer of the cortex is the endoderm. Nutrient solutions entering the central cylinder from the cortex through endodermal cells pass only through the protoplast of cells.

The bark surrounds the central cylinder of the root. It borders on a layer of cells that retain the ability to divide for a long time. This is a pericycle. Pericycle cells give rise to lateral roots, adventitious buds and secondary educational tissues. Inward from the pericycle, in the center of the root, there are conductive tissues: bast and wood. Together they form a radial conductive bundle.

The root vascular system conducts water and minerals from the root to the stem (upward current) and organic matter from the stem to the root (downward current). It consists of vascular-fibrous bundles. The main components of the bundle are sections of the phloem (through which substances move to the root) and xylem (through which substances move from the root). The main conducting elements of phloem are sieve tubes, xylem is trachea (vessels) and tracheids.

Root life processes

Transport of water in the root

Absorption of water by root hairs from the soil nutrient solution and conduction of it in a radial direction along the cells of the primary cortex through passage cells in the endoderm to the xylem of the radial vascular bundle. The intensity of water absorption by root hairs is called suction force (S), it is equal to the difference between osmotic (P) and turgor (T) pressure: S=P-T.

When the osmotic pressure is equal to the turgor pressure (P=T), then S=0, water stops flowing into the root hair cell. If the concentration of substances in the soil nutrient solution is higher than inside the cell, then water will leave the cells and plasmolysis will occur - the plants will wither. This phenomenon is observed in dry soil conditions, as well as with excessive application. mineral fertilizers. Inside the root cells, the suction force of the root increases from the rhizoderm towards the central cylinder, so water moves along a concentration gradient (i.e. from a place with a higher concentration to a place with a lower concentration) and creates root pressure, which raises the column of water through the xylem vessels , forming an ascending current. This can be found on leafless trunks in the spring when the “sap” is collected, or on cut stumps. The flow of water from wood, fresh stumps, and leaves is called “crying” of plants. When the leaves bloom, they also create a suction force and attract water to themselves - a continuous column of water is formed in each vessel - capillary tension. Root pressure is the lower driver of water flow, and the suction force of the leaves is the upper one. This can be confirmed using simple experiments.

Absorption of water by roots

Target: find out the basic function of the root.

What we do: plant grown on wet sawdust, shake off its root system and lower its roots into a glass of water. Pour a thin layer over the water to protect it from evaporation. vegetable oil and mark the level.

What we see: After a day or two, the water in the container dropped below the mark.

Result: consequently, the roots sucked up the water and brought it up to the leaves.

You can also do one more experiment to prove the absorption of nutrients by the root.

What we do: cut off the stem of the plant, leaving a stump 2-3 cm high. We put a rubber tube 3 cm long on the stump, and on the upper end we put a curved glass tube 20-25 cm high.

What we see: The water in the glass tube rises and flows out.

Result: this proves that the root absorbs water from the soil into the stem.

Does water temperature affect the intensity of water absorption by roots?

Target: find out how temperature affects root function.

What we do: one glass should be with warm water (+17-18ºС), and the other with cold water (+1-2ºС).

What we see: in the first case, water is released abundantly, in the second - little, or stops altogether.

Result: this is proof that temperature greatly influences root function.

Warm water is actively absorbed by the roots. Root pressure increases.

Cold water is poorly absorbed by the roots. In this case, root pressure drops.

Mineral nutrition

The physiological role of minerals is very great. They are the basis for the synthesis of organic compounds, as well as factors that change the physical state of colloids, i.e. directly affect the metabolism and structure of the protoplast; act as catalysts for biochemical reactions; affect cell turgor and protoplasm permeability; are centers of electrical and radioactive phenomena in plant organisms.

It has been established that normal plant development is possible only if there are three non-metals in the nutrient solution - nitrogen, phosphorus and sulfur and four metals - potassium, magnesium, calcium and iron. Each of these elements has an individual meaning and cannot be replaced by another. These are macroelements, their concentration in the plant is 10 -2 -10%. For normal plant development, microelements are needed, the concentration of which in the cell is 10 -5 -10 -3%. These are boron, cobalt, copper, zinc, manganese, molybdenum, etc. All these elements are present in the soil, but sometimes in insufficient quantities. Therefore, mineral and organic fertilizers are added to the soil.

The plant grows and develops normally if the environment surrounding the roots contains all the necessary nutrients. This environment for most plants is soil.

Breathing of roots

For normal growth and development of the plant, fresh air must be supplied to the roots. Let's check if this is true?

Target: Does the root need air?

What we do: Let's take two identical vessels with water. Place developing seedlings in each vessel. Every day we saturate the water in one of the vessels with air using a spray bottle. Pour a thin layer of vegetable oil onto the surface of the water in the second vessel, as it delays the flow of air into the water.

What we see: After some time, the plant in the second vessel will stop growing, wither, and eventually die.

Result: The death of the plant occurs due to a lack of air necessary for the root to breathe.

Root modifications

Some plants store reserve nutrients in their roots. They accumulate carbohydrates, mineral salts, vitamins and other substances. Such roots grow greatly in thickness and acquire an unusual appearance. Both the root and the stem are involved in the formation of root crops.

Roots

If reserve substances accumulate in the main root and at the base of the stem of the main shoot, root vegetables (carrots) are formed. Plants that form root crops are mostly biennials. In the first year of life, they do not bloom and accumulate a lot of nutrients in the roots. On the second, they quickly bloom, using the accumulated nutrients and forming fruits and seeds.

Root tubers

In dahlia, reserve substances accumulate in adventitious roots, forming root tubers.

Bacterial nodules

The lateral roots of clover, lupine, and alfalfa are peculiarly changed. Bacteria settle in young lateral roots, which promotes the absorption of gaseous nitrogen from the soil air. Such roots take on the appearance of nodules. Thanks to these bacteria, these plants are able to live in nitrogen-poor soils and make them more fertile.

Stilates

Ramp, which grows in the intertidal zone, develops stilted roots. They hold large leafy shoots on unstable muddy soil high above the water.

Air

U tropical plants Living on tree branches, aerial roots develop. They are often found in orchids, bromeliads, and some ferns. Aerial roots hang freely in the air, without reaching the ground and absorbing moisture from rain or dew that falls on them.

Retractors

In bulbous and corm plants, such as crocuses, among the numerous thread-like roots there are several thicker, so-called retractor roots. By contracting, such roots pull the corm deeper into the soil.

Columnar

Ficus plants develop columnar above-ground roots, or supporting roots.

Soil as a habitat for roots

Soil for plants is the medium from which it receives water and nutrients. The amount of minerals in the soil depends on the specific characteristics of the parent rock, the activity of organisms, the life activity of the plants themselves, and the type of soil.

Soil particles compete with roots for moisture, retaining it on their surface. This is the so-called bound water, which is divided into hygroscopic and film water. It is held in place by the forces of molecular attraction. The moisture available to the plant is represented by capillary water, which is concentrated in the small pores of the soil.

An antagonistic relationship develops between moisture and the air phase of the soil. The more large pores there are in the soil, the better the gas regime of these soils, the less moisture the soil retains. The most favorable water-air regime is maintained in structural soils, where water and air exist simultaneously and do not interfere with each other - water fills the capillaries inside the structural units, and air fills the large pores between them.

The nature of the interaction between plant and soil is largely related to the absorption capacity of the soil - the ability to hold or bind chemical compounds.

Soil microflora decomposes organic matter into simpler compounds and participates in the formation of soil structure. The nature of these processes depends on the type of soil, chemical composition plant residues, physiological properties of microorganisms and other factors. Soil animals take part in the formation of soil structure: annelids, insect larvae, etc.

As a result of a combination of biological and chemical processes in the soil, a complex complex of organic substances is formed, which is combined with the term “humus”.

Water culture method

What salts the plant needs, and what effect they have on its growth and development, was established through experience with aquatic crops. The water culture method is the cultivation of plants not in soil, but in an aqueous solution mineral salts. Depending on the goal of the experiment, you can exclude a particular salt from the solution, reduce or increase its content. It was found that fertilizers containing nitrogen promote plant growth, those containing phosphorus promote the rapid ripening of fruits, and those containing potassium promote the rapid outflow of organic matter from leaves to roots. In this regard, it is recommended to apply fertilizers containing nitrogen before sowing or in the first half of summer; those containing phosphorus and potassium - in the second half of summer.

Using the water culture method, it was possible to establish not only the plant’s need for macroelements, but also to clarify the role of various microelements.

Currently, there are cases where plants are grown using hydroponics and aeroponics methods.

Hydroponics is the growing of plants in containers filled with gravel. A nutrient solution containing the necessary elements is fed into the vessels from below.

Aeroponics is the air culture of plants. With this method, the root system is in the air and is automatically (several times within an hour) sprayed with a weak solution of nutrient salts.

Before calculators, students and teachers calculated square roots by hand. There are several ways to calculate the square root of a number manually. Some of them offer only an approximate solution, others give an exact answer.

Steps

Prime factorization

Factor the radical number into factors that are square numbers. Depending on the radical number, you will get an approximate or exact answer. Square numbers are numbers from which the whole square root can be taken. Factors are numbers that, when multiplied, give the original number. For example, the factors of the number 8 are 2 and 4, since 2 x 4 = 8, the numbers 25, 36, 49 are square numbers, since √25 = 5, √36 = 6, √49 = 7. Square factors are factors , which are square numbers. First, try to factor the radical number into square factors.

• For example, calculate the square root of 400 (by hand). First try factoring 400 into square factors. 400 is a multiple of 100, that is, divisible by 25 - this is a square number. Dividing 400 by 25 gives you 16. The number 16 is also a square number. Thus, 400 can be factored into the square factors of 25 and 16, that is, 25 x 16 = 400.
• This can be written as follows: √400 = √(25 x 16).

1. The square root of the product of some terms is equal to the product of the square roots of each term, that is, √(a x b) = √a x √b. Use this rule to take the square root of each square factor and multiply the results to find the answer.

   • In our example, take the root of 25 and 16.
     • √(25 x 16)
     • √25 x √16
     • 5 x 4 = 20

2. If the radical number does not factor into two square factors (and this happens in most cases), you will not be able to find the exact answer in the form of a whole number. But you can simplify the problem by decomposing the radical number into a square factor and an ordinary factor (a number from which the whole square root cannot be taken). Then you will take the square root of the square factor and will take the root of the common factor.

   • For example, calculate the square root of the number 147. The number 147 cannot be factored into two square factors, but it can be factorized into the following factors: 49 and 3. Solve the problem as follows:
     • = √(49 x 3)
     • = √49 x √3
     • = 7√3

3. If necessary, estimate the value of the root. Now you can estimate the value of the root (find an approximate value) by comparing it with the values ​​of the roots of the square numbers that are closest (on both sides of the number line) to the radical number. You will receive the root value as a decimal fraction, which must be multiplied by the number behind the root sign.

   • Let's return to our example. The radical number is 3. The square numbers closest to it will be the numbers 1 (√1 = 1) and 4 (√4 = 2). Thus, the value of √3 is located between 1 and 2. Since the value of √3 is probably closer to 2 than to 1, our estimate is: √3 = 1.7. We multiply this value by the number at the root sign: 7 x 1.7 = 11.9. If you do the math on a calculator, you'll get 12.13, which is pretty close to our answer.
     • This method also works with large numbers. For example, consider √35. The radical number is 35. The closest square numbers to it will be the numbers 25 (√25 = 5) and 36 (√36 = 6). Thus, the value of √35 is located between 5 and 6. Since the value of √35 is much closer to 6 than to 5 (because 35 is only 1 less than 36), we can say that √35 is slightly less than 6. Check on the calculator gives us the answer 5.92 - we were right.

4. Another way is to factor the radical number into prime factors. Prime factors are numbers that are divisible only by 1 and themselves. Write the prime factors in a series and find pairs of identical factors. Such factors can be taken out of the root sign.

   • For example, calculate the square root of 45. We factor the radical number into prime factors: 45 = 9 x 5, and 9 = 3 x 3. Thus, √45 = √(3 x 3 x 5). 3 can be taken out as a root sign: √45 = 3√5. Now we can estimate √5.
   • Let's look at another example: √88.
     • = √(2 x 44)
     • = √ (2 x 4 x 11)
     • = √ (2 x 2 x 2 x 11). You received three multipliers of 2; take a couple of them and move them beyond the root sign.
     • = 2√(2 x 11) = 2√2 x √11. Now you can evaluate √2 and √11 and find an approximate answer.

   Calculating square root manually

   Using long division

   1. This method involves a process similar to long division and provides an accurate answer. First, draw a vertical line dividing the sheet into two halves, and then to the right and slightly below the top edge of the sheet, draw a horizontal line to the vertical line. Now divide the radical number into pairs of numbers, starting with the fractional part after the decimal point. So, the number 79520789182.47897 is written as "7 95 20 78 91 82, 47 89 70".

      • For example, let's calculate the square root of the number 780.14. Draw two lines (as shown in the picture) and write the given number in the form “7 80, 14” at the top left. It is normal that the first digit from the left is an unpaired digit. You will write the answer (the root of this number) at the top right.

   2. For the first pair of numbers (or single number) from the left, find the largest integer n whose square is less than or equal to the pair of numbers (or single number) in question. In other words, find the square number that is closest to, but smaller than, the first pair of numbers (or single number) from the left, and take the square root of that square number; you will get the number n. Write the n you found at the top right, and write the square of n at the bottom right.

      • In our case, the first number on the left will be 7. Next, 4< 7, то есть 2 2 < 7 и n = 2. Напишите 2 сверху справа - это первая цифра в искомом квадратном корне. Напишите 2×2=4 справа снизу; вам понадобится это число для последующих вычислений.

   3. Subtract the square of the number n you just found from the first pair of numbers (or single number) on the left. Write the result of the calculation under the subtrahend (the square of the number n).

      • In our example, subtract 4 from 7 and get 3.

   4. Take down the second pair of numbers and write it down next to the value obtained in the previous step. Then double the number at the top right and write the result at the bottom right with the addition of "_×_=".

      • In our example, the second pair of numbers is "80". Write "80" after the 3. Then, double the number on the top right gives 4. Write "4_×_=" on the bottom right.

   5. Fill in the blanks on the right.

      • In our case, if we put the number 8 instead of dashes, then 48 x 8 = 384, which is more than 380. Therefore, 8 is too large a number, but 7 will do. Write 7 instead of dashes and get: 47 x 7 = 329. Write 7 at the top right - this is the second digit in the desired square root of the number 780.14.

   6. Subtract the resulting number from the current number on the left. Write the result from the previous step under the current number on the left, find the difference and write it under the subtrahend.

      • In our example, subtract 329 from 380, which equals 51.

   7. Repeat step 4. If the pair of numbers being transferred is the fractional part of the original number, then put a separator (comma) between the integer and fractional parts in the required square root at the top right. On the left, bring down the next pair of numbers. Double the number at the top right and write the result at the bottom right with the addition of "_×_=".

      • In our example, the next pair of numbers to be removed will be the fractional part of the number 780.14, so place the separator of the integer and fractional parts in the desired square root in the upper right. Take down 14 and write it in the bottom left. Double the number on the top right (27) is 54, so write "54_×_=" on the bottom right.

   8. Repeat steps 5 and 6. Find the largest number in place of the dashes on the right (instead of the dashes you need to substitute the same number) so that the result of the multiplication is less than or equal to the current number on the left.

      • In our example, 549 x 9 = 4941, which is less than the current number on the left (5114). Write 9 on the top right and subtract the result of the multiplication from the current number on the left: 5114 - 4941 = 173.

   9. If you need to find more decimal places for the square root, write a couple of zeros to the left of the current number and repeat steps 4, 5, and 6. Repeat steps until you get the answer precision (number of decimal places) you need.

   Understanding the Process

   To master this method, imagine the number whose square root you need to find as the area of ​​the square S. In this case, you will look for the length of the side L of such a square. We calculate the value of L such that L² = S.

   Give a letter for each number in the answer. Let us denote by A the first digit in the value of L (the desired square root). B will be the second digit, C the third and so on.

   Specify a letter for each pair of first digits. Let us denote by S a the first pair of digits in the value of S, by S b the second pair of digits, and so on.

   Understand the connection between this method and long division. Just like in division, where we are only interested in the next digit of the number we are dividing each time, when calculating a square root, we work through a pair of digits sequentially (to get the next one digit in the square root value).

   1. Consider the first pair of digits Sa of the number S (Sa = 7 in our example) and find its square root. In this case, the first digit A of the desired square root value will be a digit whose square is less than or equal to S a (that is, we are looking for an A such that the inequality A² ≤ Sa< (A+1)²). В нашем примере, S1 = 7, и 2² ≤ 7 < 3²; таким образом A = 2.

      • Let's say we need to divide 88962 by 7; here the first step will be similar: we consider the first digit of the divisible number 88962 (8) and select the largest number that, when multiplied by 7, gives a value less than or equal to 8. That is, we are looking for a number d for which the inequality is true: 7 × d ≤ 8< 7×(d+1). В этом случае d будет равно 1.

   2. Mentally imagine a square whose area you need to calculate. You are looking for L, that is, the length of the side of a square whose area is equal to S. A, B, C are the numbers in the number L. You can write it differently: 10A + B = L (for a two-digit number) or 100A + 10B + C = L (for three-digit number) and so on.

      • Let (10A+B)² = L² = S = 100A² + 2×10A×B + B². Remember that 10A+B is a number in which the digit B stands for units and the digit A stands for tens. For example, if A=1 and B=2, then 10A+B is equal to the number 12. (10A+B)² is the area of ​​the entire square, 100A²- area of ​​the large inner square, B²- area of ​​the small inner square, 10A×B- the area of ​​each of the two rectangles. By adding up the areas of the described figures, you will find the area of ​​the original square.

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 You can use any number without any 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 complex creative process - choosing an answer - is greatly simplified if you remember squares of popular numbers. Like a multiplication table. If, say, you need to multiply 4 by 6, you don’t add four 6 times, do you? The answer 24 immediately comes up. Although, not everyone gets it, yes...

To work freely and successfully 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 not just 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.)

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You can get acquainted with functions and derivatives.

In this article we will introduce concept of a root of a number. We will proceed sequentially: we will start with the square root, from there we will move on to the description of the cubic root, after which we will generalize the concept of a root, defining the nth root. At the same time, we will introduce definitions, notations, give examples of roots and give the necessary explanations and comments.

Square root, arithmetic square root

To understand the definition of the root of a number, and the square root in particular, you need to have . At this point we will often encounter the second power of a number - the square of a number.

Let's start with square root definitions.

Definition

Square root of a is a number whose square is equal to a.

In order to bring examples of square roots, take several numbers, for example, 5, −0.3, 0.3, 0, and square them, we get the numbers 25, 0.09, 0.09 and 0, respectively (5 2 =5·5=25, (−0.3) 2 =(−0.3)·(−0.3)=0.09, (0.3) 2 =0.3·0.3=0.09 and 0 2 =0·0=0 ). Then, by the definition given above, the number 5 is the square root of the number 25, the numbers −0.3 and 0.3 are the square roots of 0.09, and 0 is the square root of zero.

It should be noted that not for any number a there exists a whose square is equal to a. Namely, for any negative number a there is no real number b, whose square would be equal to a. In fact, the equality a=b 2 is impossible for any negative a, since b 2 is a non-negative number for any b. Thus, there is no square root of a negative number on the set of real numbers. In other words, on the set of real numbers the square root of a negative number is not defined and has no meaning.

This leads to a logical question: “Is there a square root of a for any non-negative a”? The answer is yes. This fact can be justified by the constructive method used to find the value of the square root.

Then the next logical question arises: “What is the number of all square roots of a given non-negative number a - one, two, three, or even more”? Here's the answer: if a is zero, then the only square root of zero is zero; if a is some positive number, then the number of square roots of the number a is two, and the roots are . Let's justify this.

Let's start with the case a=0 . First, let's show that zero is indeed the square root of zero. This follows from the obvious equality 0 2 =0·0=0 and the definition of the square root.

Now let's prove that 0 is the only square root of zero. Let's use the opposite method. Suppose there is some nonzero number b that is the square root of zero. Then the condition b 2 =0 must be satisfied, which is impossible, since for any non-zero b the value of the expression b 2 is positive. We have arrived at a contradiction. This proves that 0 is the only square root of zero.

Let's move on to cases where a is a positive number. We said above that there is always a square root of any non-negative number, let the square root of a be the number b. Let's say that there is a number c, which is also the square root of a. Then, by the definition of a square root, the equalities b 2 =a and c 2 =a are true, from which it follows that b 2 −c 2 =a−a=0, but since b 2 −c 2 =(b−c)·( b+c) , then (b−c)·(b+c)=0 . The resulting equality is valid properties of operations with real numbers possible only when b−c=0 or b+c=0 . Thus, the numbers b and c are equal or opposite.

If we assume that there is a number d, which is another square root of the number a, then by reasoning similar to those already given, it is proved that d is equal to the number b or the number c. So, the number of square roots of a positive number is two, and the square roots are opposite numbers.

For the convenience of working with square roots, the negative root is “separated” from the positive one. For this purpose, it is introduced definition of arithmetic square root.

Definition

Arithmetic square root of a non-negative number a is a non-negative number whose square is equal to a.

The notation for the arithmetic square root of a is . The sign is called the arithmetic square root sign. It is also called the radical sign. Therefore, you can sometimes hear both “root” and “radical”, which means the same object.

The number under the arithmetic square root sign is called radical number, and the expression under the root sign is radical expression, while the term “radical number” is often replaced by “radical expression”. For example, in the notation the number 151 is a radical number, and in the notation the expression a is a radical expression.

When reading, the word "arithmetic" is often omitted, for example, the entry is read as "the square root of seven point twenty-nine." The word “arithmetic” is used only when they want to emphasize that we are talking specifically about the positive square root of a number.

In light of the introduced notation, it follows from the definition of an arithmetic square root that for any non-negative number a .

Square roots of a positive number a are written using the arithmetic square root sign as and . For example, the square roots of 13 are and . The arithmetic square root of zero is zero, that is, . For negative numbers a, we will not attach meaning to the notation until we study complex numbers. For example, the expressions and are meaningless.

Based on the definition of the square root, the properties of square roots are proved, which are often used in practice.

In conclusion of this point, we note that the square roots of the number a are solutions of the form x 2 =a with respect to the variable x.

Cube root of a number

Definition of cube root of the number a is given similarly to the definition of the square root. Only it is based on the concept of a cube of a number, not a square.

Definition

Cube root of a is a number whose cube is equal to a.

Let's give examples of cube roots. To do this, take several numbers, for example, 7, 0, −2/3, and cube them: 7 3 =7·7·7=343, 0 3 =0·0·0=0, . Then, based on the definition of a cube root, we can say that the number 7 is the cube root of 343, 0 is the cube root of zero, and −2/3 is the cube root of −8/27.

It can be shown that the cube root of a number, unlike the square root, always exists, not only for non-negative a, but also for any real number a. To do this, you can use the same method that we mentioned when studying square roots.

Moreover, there is only a single cube root of a given number a. Let us prove the last statement. To do this, consider three cases separately: a is a positive number, a=0 and a is a negative number.

It is easy to show that if a is positive, the cube root of a can be neither a negative number nor zero. Indeed, let b be the cube root of a, then by definition we can write the equality b 3 =a. It is clear that this equality cannot be true for negative b and for b=0, since in these cases b 3 =b·b·b will be a negative number or zero, respectively. So the cube root of a positive number a is a positive number.

Now suppose that in addition to the number b there is another cube root of the number a, let's denote it c. Then c 3 =a. Therefore, b 3 −c 3 =a−a=0, but b 3 −c 3 =(b−c)·(b 2 +b·c+c 2)(this is the abbreviated multiplication formula difference of cubes), whence (b−c)·(b 2 +b·c+c 2)=0. The resulting equality is possible only when b−c=0 or b 2 +b·c+c 2 =0. From the first equality we have b=c, and the second equality has no solutions, since its left side is a positive number for any positive numbers b and c as the sum of three positive terms b 2, b·c and c 2. This proves the uniqueness of the cube root of a positive number a.

When a=0, the cube root of the number a is only the number zero. Indeed, if we assume that there is a number b, which is a non-zero cube root of zero, then the equality b 3 =0 must hold, which is possible only when b=0.

For negative a, arguments similar to the case for positive a can be given. First, we show that the cube root of a negative number cannot be equal to either a positive number or zero. Secondly, we assume that there is a second cube root of a negative number and show that it will necessarily coincide with the first.

So, there is always a cube root of any given real number a, and a unique one.

Let's give definition of arithmetic cube root.

Definition

Arithmetic cube root of a non-negative number a is a non-negative number whose cube is equal to a.

The arithmetic cube root of a non-negative number a is denoted as , the sign is called the sign of the arithmetic cube root, the number 3 in this notation is called root index. The number under the root sign is radical number, the expression under the root sign is radical expression.

Although the arithmetic cube root is defined only for non-negative numbers a, it is also convenient to use notations in which negative numbers are found under the arithmetic cube root sign. We will understand them as follows: , where a is a positive number. For example, .

We will talk about the properties of cube roots in general article properties of roots.

Calculating the value of a cube root is called extracting a cube root; this action is discussed in the article extracting roots: methods, examples, solutions.

To conclude this point, let's say that the cube root of the number a is a solution of the form x 3 =a.

nth root, arithmetic root of degree n

Let us generalize the concept of a root of a number - we introduce definition of nth root for n.

Definition

nth root of a is a number whose nth power is equal to a.

From this definition it is clear that the first degree root of the number a is the number a itself, since when studying the degree with a natural exponent we took a 1 =a.

Above we looked at special cases of the nth root for n=2 and n=3 - square root and cube root. That is, a square root is a root of the second degree, and a cube root is a root of the third degree. To study roots of the nth degree for n=4, 5, 6, ..., it is convenient to divide them into two groups: the first group - roots of even degrees (that is, for n = 4, 6, 8, ...), the second group - roots odd degrees (that is, with n=5, 7, 9, ...). This is due to the fact that roots of even powers are similar to square roots, and roots of odd powers are similar to cubic roots. Let's deal with them one by one.

Let's start with the roots whose powers are the even numbers 4, 6, 8, ... As we already said, they are similar to the square root of the number a. That is, the root of any even degree of the number a exists only for non-negative a. Moreover, if a=0, then the root of a is unique and equal to zero, and if a>0, then there are two roots of even degree of the number a, and they are opposite numbers.

Let us substantiate the last statement. Let b be an even root (we denote it as 2·m, where m is some natural number) of the number a. Suppose that there is a number c - another root of degree 2·m from the number a. Then b 2·m −c 2·m =a−a=0 . But we know the form b 2 m −c 2 m = (b−c) (b+c) (b 2 m−2 +b 2 m−4 c 2 +b 2 m−6 c 4 +…+c 2 m−2), then (b−c)·(b+c)· (b 2 m−2 +b 2 m−4 c 2 +b 2 m−6 c 4 +…+c 2 m−2)=0. From this equality it follows that b−c=0, or b+c=0, or b 2 m−2 +b 2 m−4 c 2 +b 2 m−6 c 4 +…+c 2 m−2 =0. The first two equalities mean that the numbers b and c are equal or b and c are opposite. And the last equality is valid only for b=c=0, since on its left side there is an expression that is non-negative for any b and c as the sum of non-negative numbers.

As for the roots of the nth degree for odd n, they are similar to the cube root. That is, the root of any odd degree of the number a exists for any real number a, and for a given number a it is unique.

The uniqueness of a root of odd degree 2·m+1 of the number a is proved by analogy with the proof of the uniqueness of the cube root of a. Only here instead of equality a 3 −b 3 =(a−b)·(a 2 +a·b+c 2) an equality of the form b 2 m+1 −c 2 m+1 = is used (b−c)·(b 2·m +b 2·m−1 ·c+b 2·m−2 ·c 2 +… +c 2·m). The expression in the last bracket can be rewritten as b 2 m +c 2 m +b c (b 2 m−2 +c 2 m−2 + b c (b 2 m−4 +c 2 m−4 +b c (…+(b 2 +c 2 +b c)))). For example, with m=2 we have b 5 −c 5 =(b−c)·(b 4 +b 3 ·c+b 2 ·c 2 +b·c 3 +c 4)= (b−c)·(b 4 +c 4 +b·c·(b 2 +c 2 +b·c)). When a and b are both positive or both negative, their product is a positive number, then the expression b 2 +c 2 +b·c in the highest nested parentheses is positive as the sum of the positive numbers. Now, moving sequentially to the expressions in brackets of the previous degrees of nesting, we are convinced that they are also positive as the sum of positive numbers. As a result, we obtain that the equality b 2 m+1 −c 2 m+1 = (b−c)·(b 2·m +b 2·m−1 ·c+b 2·m−2 ·c 2 +… +c 2·m)=0 possible only when b−c=0, that is, when the number b is equal to the number c.

It's time to understand the notation of nth roots. For this purpose it is given definition of arithmetic root of the nth degree.

Definition

Arithmetic root of the nth degree of a non-negative number a is a non-negative number whose nth power is equal to a.