How to find a fractional root. Square root. An exhaustive guide (2019). Root word: definition of the concept

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

Steps

Prime factorization

Factor the root number into factors that are square numbers. Depending on the root number, you will get an approximate or exact answer. Square numbers - numbers from which you can extract an integer Square root. 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 factorize the root number into square factors.

For example, calculate the square root of 400 (manually). 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 square factors of 25 and 16, that is, 25 x 16 = 400.
This can be written as follows: √400 = √(25 x 16).

Square root of the product of some terms is equal to the product square roots from each term, i.e. √(a x b) = √a x √b. Use this rule and take the square root of each square factor and multiply the results to find the answer.
- In our example, take the square root of 25 and 16.
  - √(25 x 16)
  - √25 x √16
  - 5 x 4 = 20
If the radical number does not decompose into two square multiplier(which happens most of the time), you won't be able to find the exact answer as an integer. But you can simplify the problem by decomposing the root 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 you will take the root of the ordinary 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 factored into the following factors: 49 and 3. Solve the problem as follows:
  - = √(49 x 3)
  - = √49 x √3
  - = 7√3
If necessary, evaluate the value of the root. Now you can evaluate the value of the root (find an approximate value) by comparing it with the values of the roots of square numbers that are closest (on both sides of the number line) to the root number. You will get the value of the root as a decimal fraction, which must be multiplied by the number behind the root sign.
- Let's go back to our example. The root number is 3. The nearest square numbers to it are the numbers 1 (√1 = 1) and 4 (√4 = 2). Thus, the value of √3 lies 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 \u003d 11.9. If you do the calculations on a calculator, you get 12.13, which is pretty close to our answer.
  - This method also works with big numbers. For example, consider √35. The root number is 35. The nearest square numbers to it are the numbers 25 (√25 = 5) and 36 (√36 = 6). Thus, the value of √35 lies between 5 and 6. Since the value of √35 is much closer to 6 than it is to 5 (because 35 is only 1 less than 36), we can state that √35 is slightly less than 6. Checking with a calculator gives us the answer 5.92 - we were right.
Another way is to decompose the root number into prime factors. Prime factors are numbers that are only divisible by 1 and themselves. write down prime factors in a row and find pairs of identical factors. Such factors can be taken out of the sign of the root.
- For example, calculate the square root of 45. We decompose the root number into prime factors: 45 \u003d 9 x 5, and 9 \u003d 3 x 3. Thus, √45 \u003d √ (3 x 3 x 5). 3 can be taken out of the root sign: √45 = 3√5. Now we can estimate √5.
- Consider another example: √88.
  - = √(2 x 44)
  - = √ (2 x 4 x 11)
  - = √ (2 x 2 x 2 x 11). You got three multiplier 2s; take a couple of them and take them out of the sign of the root.
  - = 2√(2 x 11) = 2√2 x √11. Now we can evaluate √2 and √11 and find an approximate answer.
Calculating the square root manually

Using column division
1. This method involves a process similar to long division and gives an accurate answer. First, draw a vertical line dividing the sheet into two halves, and then draw a horizontal line to the right and slightly below the top edge of the sheet to the vertical line. Now divide the root 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 number in the top left as "7 80, 14". It is normal that the first digit from the left is an unpaired digit. The answer (the root of the given number) will be written on the top right.
2. Given the first pair of numbers (or one number) from the left, find the largest integer n whose square is less than or equal to the pair of numbers (or one number) in question. In other words, find the square number that is closest to, but less 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 found n at the top right, and write down the square n at the bottom right.
  - In our case, the first number on the left will be the number 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 one number) from the left. Write the result of the calculation under the subtrahend (the square of the number n).
  - In our example, subtract 4 from 7 to 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 "_×_=" appended.
  - In our example, the second pair of numbers is "80". Write "80" after the 3. Then, doubling the number from the top right gives 4. Write "4_×_=" from the bottom right.
5. Fill in the blanks on the right.
  - In our case, if instead of dashes we put the number 8, then 48 x 8 \u003d 384, which is more than 380. Therefore, 8 is too large a number, but 7 is fine. Write 7 instead of dashes and get: 47 x 7 \u003d 329. Write 7 from 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 below the current number on the left, find the difference and write it below the subtracted one.
  - In our example, subtract 329 from 380, which equals 51.
7. Repeat step 4. If the demolished pair of numbers is the fractional part of the original number, then put the separator (comma) of the integer and fractional parts in the desired square root from the top right. On the left, carry down the next pair of numbers. Double the number at the top right and write the result at the bottom right with "_×_=" appended.
  - In our example, the next pair of numbers to be demolished will be the fractional part of the number 780.14, so put the separator of the integer and fractional parts in the desired square root from the top right. Demolish 14 and write down at the bottom left. Double the top right (27) is 54, so write "54_×_=" at the bottom right.
8. Repeat steps 5 and 6. Find the largest number in place of dashes on the right (instead of dashes you need to substitute the same number) so that the multiplication result 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 pair of zeros next to the current number on the left and repeat steps 4, 5 and 6. Repeat steps until you get the accuracy of the answer you need (number of decimal places).
Understanding the process
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 sought value of the square root will be such a digit, the square of which is less than or equal to S a (that is, we are looking for such an A that satisfies 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 the 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 S. A, B, C are numbers in the number L. You can write it differently: 10A + B \u003d L (for a two-digit number) or 100A + 10B + C \u003d 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 whose B stands for ones and A stands for tens. For example, if A=1 and B=2, then 10A+B equals the number 12. (10A+B)² is the area of the whole square, 100A² is the area of the large inner square, B² is the area of the small inner square, 10A×B is the area of each of the two rectangles. Adding the areas of the figures described, you will find the area of the original square.

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 place of deposition in the supply of nutrients. In this case, it is modified.

Root types

The main root is formed from the germinal root during seed germination. It has lateral roots.

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 contributes to a better strengthening of the plant in the soil and improves its nutrition.

Types of root systems

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

A 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 surface of the roots is 200 m 2.

This is many times greater than the area of the above-ground mass.

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

External structure of the root. The internal structure of the root

Root zones

root cap

The root grows in length with its tip, where the young cells of the educational tissue are located. The growing part is covered with a root cap that protects the tip of the root 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 to be covered with mucus, which reduces friction between the root and soil particles. They can even push apart soil particles. The cells of the root cap are living, often containing grains of starch. The cells of the cap are constantly updated due to division. Participates in positive geotropical reactions (direction of root growth towards the center of the Earth).

The cells of the division zone are actively dividing, the length of this zone varies in different species and in different roots of the same plant.

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

As linear growth is completed, the third stage of root formation begins - its differentiation, a zone of differentiation and specialization of cells (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.

The structure of the root hair

Root hairs are highly elongated outgrowths of the outer cells covering the root. The number of root hairs is very high (from 200 to 300 hairs per 1 mm2). Their length reaches 10 mm. Hairs are formed very quickly (in young seedlings of an apple tree in 30-40 hours). Root hairs are short-lived. They die off in 10-20 days, and new ones grow on the young part of the root. This ensures the development of new soil horizons by the root. 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 off is able to absorb water for some time, but then becomes covered with cork and loses this ability.

The sheath of the hair 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 gluing of root hairs with 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 a support for the top of the root, which passes between the soil particles.

Under a microscope, on a cross section of the root in the absorption zone, its structure is visible on the cellular and tissue levels. On the surface of the root is the rhizoderm, below it is the bark. The outer layer of the cortex is the exoderm, inward from it is the main parenchyma. Its thin-walled living cells perform a storage function, conduct nutrient solutions in the radial direction - from the absorbing tissue to the vessels of the wood. They also synthesize a number of vital for the plant organic matter. The inner layer of the cortex is the endoderm. Nutrient solutions coming from the cortex to the central cylinder through the cells of the endoderm pass only through the protoplast of the 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 the pericycle. Pericycle cells give rise to lateral roots, adnexal 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 conducting beam.

The conducting system of the root 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 the phloem are sieve tubes, xylems are tracheas (vessels) and tracheids.

Root life processes

Water transport at the root

Absorption of water by root hairs from the soil nutrient solution and its conduction in the radial direction along the cells of the primary cortex through the passage cells in the endodermis to the xylem of the radial vascular bundle. The intensity of water absorption by the root hairs is called the suction force (S), it is equal to the difference between the 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 conditions of dry soil, as well as with excessive application of mineral fertilizers. Inside the root cells, the sucking force of the root increases from the rhizoderm towards the central cylinder, so water moves along the concentration gradient (i.e., from a place with a higher concentration to a place with a lower concentration) and creates a root pressure that raises the column of water along the xylem vessels , forming an upward current. It can be found on spring leafless trunks when "sap" is harvested, or on cut stumps. The outflow of water from wood, fresh stumps, leaves, is called the "weeping" of plants. When the leaves bloom, they also create a sucking force and attract water to themselves - a continuous column of water is formed in each vessel - capillary tension. Root pressure is the lower motor of the water current, and the sucking power of the leaves is the upper one. You can confirm this with the help of simple experiments.

Absorption of water by roots

Target: find out the main function of the root.

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

What we observe: after a day or two, the water in the tank dropped below the mark.

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

One more experiment can be done, proving the absorption of nutrients by the root.

What we do: we 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 put a curved glass tube 20-25 cm high on the upper end.

What we observe: 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 the temperature of the water affect the rate of absorption of water by the root?

Target: find out how temperature affects root operation.

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

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

Result: this is proof that temperature has a strong effect on root performance.

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

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

mineral nutrition

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

It has been established that the normal development of plants is possible only in the presence of three non-metals in the nutrient solution - nitrogen, phosphorus and sulfur and - and four metals - potassium, magnesium, calcium and iron. Each of these elements has individual value and cannot be replaced by another. These are macronutrients, their concentration in the plant is 10 -2 -10%. For the normal development of plants, 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 found in the soil, but sometimes in insufficient quantities. Therefore, mineral and organic fertilizers are applied to the soil.

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

Root breath

For normal growth and development of the plant, it is necessary that fresh air enter the root. Let's check if it is?

Target: do roots need air?

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

What we observe: after a while, the plant in the second vessel will stop growing, wither, and eventually die.

Result: the death of the plant occurs due to the lack of air necessary for the respiration of the root.

Root modifications

In some plants, reserve nutrients are deposited in the roots. They accumulate carbohydrates, mineral salts, vitamins and other substances. Such roots grow strongly 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 crops (carrots) are formed. Root-forming plants are mostly biennials. In the first year of life, they do not bloom and accumulate a lot of nutrients in root crops. On the second, they quickly bloom, using the accumulated nutrients and form fruits and seeds.

root tubers

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

bacterial nodules

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

stilted

A ramp growing in the intertidal zone develops stilted roots. High above the water, they hold large leafy shoots on unsteady muddy ground.

Air

Tropical plants that live on tree branches develop aerial roots. They are often found in orchids, bromeliads, and some ferns. Aerial roots hang freely in the air, not reaching the ground and absorbing moisture from rain or dew that falls on them.

Retractors

In bulbous and tuber-bulbous plants, for example, in crocuses, among numerous filamentous roots, there are several thicker, so-called retracting roots. Reducing, such roots draw the corm deeper into the soil.

Pillar-shaped

Ficus develop columnar above-ground roots, or support roots.

Soil as a habitat for roots

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

Soil particles compete with roots for moisture, holding it on their surface. This is the so-called bound water, which is divided into hygroscopic and film. It is held 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.

Antagonistic relations develop between the moisture and the air phase of the soil. The more large pores 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 are located simultaneously and do not interfere with each other - water fills the capillaries inside the structural aggregates, and air fills the large pores between them.

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

Soil microflora decomposes organic matter into simpler compounds, participates in the formation of soil structure. The nature of these processes depends on the type of soil, chemical composition plant remains, physiological properties 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 a complex complex of organic substances is formed in the soil, which is combined by the term "humus".

Water culture method

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

Using the method of water cultures, it was possible to establish not only the need of a plant for macroelements, but also to find out the role of various microelements.

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

Hydroponics is the cultivation of plants in pots filled with gravel. Nutrient solution containing 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 automatically (several times within an hour) is sprayed with a weak solution of nutrient salts.

Root formulas. properties of square roots.

Attention!
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material in Special Section 555.
For those who strongly "not very..."
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In the previous lesson, we figured out what a square root is. It's time to figure out what are formulas for roots, what are root properties and what can be done about it all.

Root Formulas, Root Properties, and Rules for Actions with Roots- it's essentially the same thing. There are surprisingly few formulas for square roots. Which, of course, pleases! Rather, you can write a lot of all sorts of formulas, but only three are enough for practical and confident work with roots. Everything else flows from these three. Although many stray in the three formulas of the roots, yes ...

Let's start with the simplest. There she is:

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

Students always ask: “Why can't I use a calculator on a math exam? How to extract the square root of a number without a calculator? Let's try to answer this question.

How to extract the square root of a number without the help of a calculator?

Action square root extraction the opposite of squaring.

√81= 9 9 2 =81

If from positive number take the square root and square the result, we get the same number.

From small numbers that are perfect squares natural numbers, for example 1, 4, 9, 16, 25, ..., 100 square roots can be extracted verbally. Usually at school they teach a table of squares of natural numbers up to twenty. Knowing this table, it is easy to extract the square roots from the numbers 121,144, 169, 196, 225, 256, 289, 324, 361, 400. From numbers greater than 400, you can extract using the selection method using some tips. Let's try an example to consider this method.

Example: Extract the root of the number 676.

We notice that 20 2 \u003d 400, and 30 2 \u003d 900, which means 20< √676 < 900.

Exact squares of natural numbers end in 0; one; four; 5; 6; 9.
The number 6 is given by 4 2 and 6 2 .
So, if the root is taken from 676, then it is either 24 or 26.

It remains to check: 24 2 = 576, 26 2 = 676.

Answer: √676 = 26 .

More example: √6889 .

Since 80 2 \u003d 6400, and 90 2 \u003d 8100, then 80< √6889 < 90.
The number 9 is given by 3 2 and 7 2, then √6889 is either 83 or 87.

Check: 83 2 = 6889.

Answer: √6889 = 83 .

If you find it difficult to solve by the selection method, then you can factorize the root expression.

For example, find √893025.

Let's factorize the number 893025, remember, you did it in the sixth grade.

We get: √893025 = √3 6 ∙5 2 ∙7 2 = 3 3 ∙5 ∙7 = 945.

More example: √20736. Let's factorize the number 20736:

We get √20736 = √2 8 ∙3 4 = 2 4 ∙3 2 = 144.

Of course, factoring requires knowledge of divisibility criteria and factoring skills.

And finally, there is square root rule. Let's look at this rule with an example.

Calculate √279841.

To extract the root of a multi-digit integer, we split it from right to left into faces containing 2 digits each (there may be one digit in the left extreme face). Write like this 27'98'41

To get the first digit of the root (5), we extract the square root of the largest exact square contained in the first left face (27).
Then the square of the first digit of the root (25) is subtracted from the first face and the next face (98) is attributed (demolished) to the difference.
To the left of the received number 298, they write the double digit of the root (10), divide by it the number of all tens of the previously obtained number (29/2 ≈ 2), experience the quotient (102 ∙ 2 = 204 should not be more than 298) and write (2) after the first digit of the root.
Then the resulting quotient 204 is subtracted from 298, and the next facet (41) is attributed (demolished) to the difference (94).
To the left of the resulting number 9441, they write the double product of the digits of the root (52 ∙ 2 = 104), divide by this product the number of all tens of the number 9441 (944/104 ≈ 9), experience the quotient (1049 ∙ 9 = 9441) should be 9441 and write it down (9) after the second digit of the root.

We got the answer √279841 = 529.

Similarly extract roots of decimals. Only the radical number must be divided into faces so that the comma is between the faces.

Example. Find the value √0.00956484.

You just have to remember that if decimal It has odd number decimal places, the exact square root is not extracted from it.

So, now you have seen three ways to extract the root. Choose the one that suits you best and practice. To learn how to solve problems, you need to solve them. And if you have any questions, sign up for my lessons.

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Language is our teacher. And every word is a lesson. The lessons of single-root words are especially interesting. Here is the tractor driver. He drives a tractor. Plantain grass grows along the road. A winter hut is a place where they spend the winter. Single-root words help to understand how the word was formed, what it means. About this in the lesson “The root of the word. One-word words. During the lesson, you will observe word families, find out what single-root words are, what is called the root of a word, make sure that the root in related words is spelled the same, and also observe the alternation of consonants in the root.

Scientists have calculated that there are approximately 4,500 roots in the Russian language. The author M.A. Rybnikova believed: “Finding the root of a word means finding its inner, hidden meaning - the same as lighting a flame inside a lantern.” Lesson topic: “The root of the word. One-word words. Writing the root in single-root words.

Some words are said to be related. Let's remember what this name means.

Related words are words that can be explained using the same word. Part of this word lives in all related words. Therefore, related words there isa common part and general meaning.

For example, sugar bowl, sugar, candy- related words?

1. Let's see if there is a common part in the words ? (The words sugar bowl, sugar have a common part sugar)

2. Is there a general meaning? (Is it possible to explain the words using the same word?)

A sugar bowl is a piece of teaware for sugar. So, sugar bowl, sugar are related words. Candy is not a related word.

Words are given: fish, fish, catch, fish, fish, perch, fisherman.

Let's collect a family of related words.

How to recognize them? Firstly, there is a common part in words (fish), and secondly, there is a common meaning. You can explain words using the same word.

Fishing - fishing. A fish is a small fish. Fish - cooked from fish. A fisherman is one who catches fish.

Means, fish, fish, small fish, fishy, fisherman- related words.

We have words catch and perch.

We select only those words that we consider related to them. Okunyok, dipped, catch, dexterous - related words?

Do the words have a common part? (Perch, fishing)

Can words be explained using the same word? A perch is a small perch. So perch and perch are related words.

Dipped - immersed in liquid. Perch, dipped - these words have no common meaning.

Catch - the number of fish caught. So, catch, catch - these are related words.

Agile - skillful, possessing physical dexterity. Catch, dexterous - these words have no common meaning.

What is the name of the common part of related words?

The common part of related words is called the root.

The root contains a common meaning for all related words.

Note the root in related words. In words perch, perch perch root. In words catch, catch lov root.

Related words are called cognates because they have the same root.

Conclusion: vowels and consonants are different.

Are the letters the same? The letters are the same.

Remember the secret of the roots! The roots of related words are spelled the same.

To find a root in a word, you need:

1. Pick up related words. 2. Select the same part.

Let's find the root in words gift, shout out, silver.

A gift is a thing that is given, brought as a gift. The common part is a gift.

Shout out - shout out loudly, shout out. The root is a cry.

Silver - the color of silver, with a silver tint. Root - silver.

By the way snow choose related words. We recognize them by the description of the value.

1. An affectionate name for snow (snowball).

2. Crystal of snow (snowflake).

3. Snowman (snowman).

4. Abundant snow (snowy).

5. Small, tightly rolled lumps of snow (snowballs).

These words have a common meaning. Let's look at the root.

Imagine that in all these words the root snow. Say each word with this root. Did you feel comfortable pronouncing "snowy", "snowy"?

You observed the law of the language: in the root of words with the same root, some consonants can be replaced by others. This substitution is called consonant alternation.

In these words, the root is snow-snow, at the root there is an alternation of consonant letters Mrs.

What other letters of consonants alternate in the root of single-root words?

Look at the last letter of the consonant in the root.

fluff-push OK

wow about- ush ko x-sh

waters it- rein ak

glance et- looking at Dr.

rivers a- rech ka

agony a- much noah k-h

the weight s-vz vesh willow

braid a- kosh at s-sh

WHO it -vozh at

tale-say and h

And in words ice-ice yana, ate point- spruce letter yo replaces a letter e.

Note! The root is considered the same, and the words are related if the letters e and yo, G and f, d-f, k-h, x-sh and others replace each other.

somehow

Many years ago

planted strange garden.

It wasn't fruity

He was only a word.

This word,

root word,

It soon began to grow

And brought us fruit -

There are many new words.

Here from the garden

You seedlings

Here are some more landings.

But

Gardener .

The gardener is with him.

Very interesting

Walk in the garden of words!

(E. Izmailov)

One-word words: garden, planted, seedling, planting, grower(Gardening Specialist) , gardener(worker who takes care of the garden).

Is it possible to add words garden, plant, soot, seedlings?

Garden- pertaining to the garden.

Plant- the same as planting.

seedlings- plants transplanted from another place. At the root of single-root words, there is an alternation of consonants dr.

But soot is of no general importance. Soot is black deposits from combustion.

Let's name a family of single-root words with the root UCH-: teacher, student, training, scientist, retrain, memorize, teacher, educational, teacher's, head teacher, teach, study.

In the lesson, you learned that the common part of related words is called the root. The roots of related words are spelled the same. Single-root words are words that have the same root and the same meaning. To find a root in a word, you need to pick up related words and highlight the same part in them.

M.S.Soloveichik, N.S. Kuzmenko "To the secrets of our language" Russian language: Textbook. Grade 3: in 2 parts. Smolensk: Association XXI century, 2010.
M.S. Soloveichik, N.S. Kuzmenko "To the secrets of our language" Russian language: Workbook. Grade 3: in 3 parts. Smolensk: Association XXI century, 2010.
T. V. Koreshkova Test tasks In Russian. Grade 3: in 2 parts. - Smolensk: Association XXI century, 2011.
T. V. Koreshkova Practice! Notebook for independent work in Russian for grade 3: in 2 parts. - Smolensk: Association XXI century, 2011.
L.V. Mashevskaya, L.V. Danbitskaya Creative tasks in the Russian language. - St. Petersburg: KARO, 2003
G.T Dyachkova Olympiad tasks in Russian. 3-4 classes. - Volgograd: Teacher, 2008

School-collection.edu.ru ().
The festival pedagogical ideas "Public lesson" ().
padabum.com ().

Write down the word salt and add the same-root words to it. Recognize them by the description of the meaning.

1) A small vessel for table salt - ...

2) Put salt in something for taste - ...

3) Possessing the taste of salt - ...

Write out from proverbs and sayings the same root words. Select the root.

1) Truth is not friends with lies.

2) In a friendly team, things are arguing.

3) I read a book - I met a friend.

4) Learn to cherish friendship.

Divide the words into two groups of single-root words.

Water, water, driver, flood, seeing off, conductor, watery, watery, guide.