The nth root of a number x is a non-negative number z that, when raised to the nth power, becomes x. Determining the root is included in the list of basic arithmetic operations that we become familiar with in childhood.

Mathematical notation

"Root" comes from the Latin word radix and today the word "radical" is used as a synonym for this mathematical term. Since the 13th century, mathematicians have denoted the root operation by the letter r with a horizontal bar over the radical expression. In the 16th century, the designation V was introduced, which gradually replaced the sign r, but the horizontal line remained. It is easy to type in a printing house or write by hand, but in electronic publications and programming, the letter designation for the root has become widespread - sqrt. This is how we will denote square roots in this article.

Square root

The square radical of a number x is a number z that, when multiplied by itself, becomes x. For example, if we multiply 2 by 2, we get 4. In this case, two is square root out of four. Multiply 5 by 5, we get 25 and now we already know the value of the expression sqrt(25). We can multiply and – 12 by −12 to get 144, and the radical of 144 is both 12 and −12. Obviously, square roots can be both positive and negative numbers.

The peculiar dualism of such roots is important for solving quadratic equations, therefore, when searching for answers in such problems, you need to indicate both roots. When deciding algebraic expressions Arithmetic square roots are used, that is, only their positive values.

Numbers whose square roots are integers are called perfect squares. There is a whole sequence of such numbers, the beginning of which looks like:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256…

The square roots of other numbers are irrational numbers. For example, sqrt(3) = 1.73205080757... and so on. This number is infinite and non-periodic, which causes some difficulties in calculating such radicals.

The school mathematics course states that you cannot take square roots of negative numbers. As we learn in a university course on mathematical analysis, this can and should be done - this is why complex numbers are needed. However, our program is designed to extract real root values, so it does not calculate even radicals from negative numbers.

Cube root

The cubic radical of a number x is a number z that, when multiplied by itself three times, gives the number x. For example, if we multiply 2 × 2 × 2, we get 8. Therefore, two is the cube root of eight. Multiply the four by itself three times and get 4 × 4 × 4 = 64. Obviously, the four is the cube root of the number 64. There is an infinite sequence of numbers whose cubic radicals are integers. Its beginning looks like:

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744…

For other numbers, the cube roots are irrational numbers. Unlike square radicals, cube roots, like any odd roots, can be derived from negative numbers. It's all about the product of numbers less than zero. Minus for minus gives a plus - a rule known from school. And a minus for a plus gives a minus. If you multiply negative numbers an odd number of times, the result will also be negative, therefore, extract the odd radical from negative number nothing bothers us.

However, the calculator program works differently. Essentially, root extraction is the construction of reciprocal degree. The square root is considered to be raised to the power of 1/2, and the cubic root is considered to be raised to the power of 1/3. The formula for raising to the power of 1/3 can be rearranged and expressed as 2/6. The result is the same, but you cannot extract such a root from a negative number. Thus, our calculator calculates arithmetic roots only from positive numbers.

nth root

Such an ornate method of calculating radicals allows you to determine roots of any degree from any expression. You can take the fifth root of a cube of a number or the 19th radical of a number to the 12th power. All this is elegantly implemented in the form of raising to the power of 3/5 or 12/19, respectively.

Let's look at an example

Diagonal of a square

The irrationality of the diagonal of a square was known to the ancient Greeks. They were faced with the problem of calculating the diagonal of a flat square, since its length is always proportional to the root of two. The formula for determining the length of the diagonal is derived from and ultimately takes the form:

d = a × sqrt(2).

Let's determine the square radical of two using our calculator. Let’s enter the value 2 in the “Number(x)” cell, and also 2 in the “Degree(n)” cell. As a result, we get the expression sqrt(2) = 1.4142. Thus, to roughly estimate the diagonal of a square, it is enough to multiply its side by 1.4142.

Conclusion

Finding a radical is a standard arithmetic operation, without which scientific or design calculations are indispensable. Of course, we don’t need to determine roots to solve everyday problems, but our online calculator will definitely be useful for schoolchildren or students to check homework in algebra or calculus.

If you have a calculator at hand, extracting the cube root of any number will not be any problem. But if you don't have a calculator or you just want to impress others, find the cube root by hand. Most people will find the process described here quite complicated, but with practice, extracting cube roots will become much easier. Before you start reading this article, remember the basic mathematical operations and calculations with cubed numbers.

Steps

Part 1

Write down the task. Taking cube roots by hand is similar to long division, but with some nuances. First, write down the task in a specific form.

• Write down the number from which you want to take the cube root. Divide the number into groups of three digits, starting with the decimal point. For example, you need to take the cube root of 10. Write this number like this: 10,000,000. The additional zeros are intended to increase the accuracy of the result.
• Draw a root sign next to and above the number. Think of it as the horizontal and vertical lines you draw when dividing. The only difference is the shape of the two signs.
• Place a decimal point above the horizontal line. Do this directly above the decimal point of the original number.

1. Remember the results of cubed integers. They will be used in calculations.

   • 1 3 = 1 ∗ 1 ∗ 1 = 1 (\displaystyle 1^(3)=1*1*1=1)
   • 2 3 = 2 ∗ 2 ∗ 2 = 8 (\displaystyle 2^(3)=2*2*2=8)
   • 3 3 = 3 ∗ 3 ∗ 3 = 27 (\displaystyle 3^(3)=3*3*3=27)
   • 4 3 = 4 ∗ 4 ∗ 4 = 64 (\displaystyle 4^(3)=4*4*4=64)
   • 5 3 = 5 ∗ 5 ∗ 5 = 125 (\displaystyle 5^(3)=5*5*5=125)
   • 6 3 = 6 ∗ 6 ∗ 6 = 216 (\displaystyle 6^(3)=6*6*6=216)
   • 7 3 = 7 ∗ 7 ∗ 7 = 343 (\displaystyle 7^(3)=7*7*7=343)
   • 8 3 = 8 ∗ 8 ∗ 8 = 512 (\displaystyle 8^(3)=8*8*8=512)
   • 9 3 = 9 ∗ 9 ∗ 9 = 729 (\displaystyle 9^(3)=9*9*9=729)
   • 10 3 = 10 ∗ 10 ∗ 10 = 1000 (\displaystyle 10^(3)=10*10*10=1000)

2. Find the first digit of the answer. Choose the cube of the integer that is closest but smaller than the first group of three digits.

   • In our example, the first group of three digits is the number 10. Find the largest cube that is less than 10. This cube is 8, and the cube root of 8 is 2.
   • Above the horizontal line above the number 10, write the number 2. Then write down the value of the operation 2 3 (\displaystyle 2^(3))= 8 under 10. Draw a line and subtract 8 from 10 (as with regular long division). The result is 2 (this is the first remainder).
   • Thus, you have found the first digit of the answer. Think about whether this result quite accurate. In most cases this will be a very rough answer. Cube the result to find out how close it is to the original number. In our example: 2 3 (\displaystyle 2^(3))= 8, which is not very close to 10, so the calculations need to be continued.

3. Find the next digit of the answer. Add a second group of three digits to the first remainder, and draw a vertical line to the left of the resulting number. Using the resulting number you will find the second digit of the answer. In our example, we need to add a second group of three digits (000) to the first remainder (2) to get the number 2000.

   • To the left of the vertical line you will write three numbers, the sum of which is equal to a certain first factor. Leave empty spaces for these numbers and put plus signs between them.

4. Find the first term (out of three). In the first empty space, write the result of multiplying the number 300 by the square of the first digit of the answer (it is written above the root sign). In our example, the first digit of the answer is 2, so 300*(2^2) = 300*4 = 1200. Write 1200 in the first blank space. The first term is the number 1200 (plus two more numbers to find).

   Find the second digit of the answer. Find out what number you need to multiply 1200 by so that the result is close, but does not exceed 2000. This number can only be 1, since 2 * 1200 = 2400, which is more than 2000. Write 1 (the second digit of the answer) after 2 and the decimal point above the root sign.

   Find the second and third terms (out of three). The multiplier consists of three numbers (terms), the first of which you have already found (1200). Now we need to find the remaining two terms.

   • Multiply 3 by 10 and by each digit of the answer (they are written above the root sign). In our example: 3*10*2*1 = 60. Add this result to 1200 and get 1260.
   • Finally, square the last digit of your answer. In our example, the last digit of the answer is 1, so 1^2 = 1. Thus, the first factor is equal to the sum of the following numbers: 1200 + 60 + 1 = 1261. Write this number to the left of the vertical bar.

5. Multiply and subtract. Multiply the last digit of the answer (in our example it is 1) by the found factor (1261): 1*1261 = 1261. Write this number under 2000 and subtract it from 2000. You will get 739 (this is the second remainder).

6. Consider whether the answer you receive is accurate enough. Do this every time you complete another subtraction. After the first subtraction, the answer was 2, which is not an accurate result. After the second subtraction, the answer is 2.1.

   • To check the accuracy of your answer, cube it: 2.1*2.1*2.1 = 9.261.
   • If you think the answer is accurate enough, you don't have to continue the calculations; otherwise, do another subtraction.

7. Find the second factor. To practice your calculations and get a more accurate result, repeat the steps above.

   • To the second remainder (739) add the third group of three digits (000). You will get the number 739000.
   • Multiply 300 by the square of the number written above the root sign (21): 300 ∗ 21 2 (\displaystyle 300*21^(2)) = 132300.
   • Find the third digit of the answer. Find out what number you need to multiply 132300 by so that the result is close to, but does not exceed 739000. This number is 5: 5 * 132200 = 661500. Write 5 (the third digit of the answer) after the 1 above the root sign.
   • Multiply 3 by 10 by 21 and by the last digit of the answer (they are written above the root sign). In our example: 3 ∗ 21 ∗ 5 ∗ 10 = 3150 (\displaystyle 3*21*5*10=3150).
   • Finally, square the last digit of your answer. In our example, the last digit of the answer is 5, so 5 2 = 25. (\displaystyle 5^(2)=25.)
   • Thus, the second multiplier is: 132300 + 3150 + 25 = 135475.

8. Multiply the last digit of the answer by the second factor. Once you have found the second factor and third digit of the answer, proceed as follows:

   • Multiply the last digit of the answer by the found factor: 135475*5 = 677375.
   • Subtract: 739000-677375 = 61625.
   • Consider whether the answer you receive is accurate enough. To do this, cube it: 2 , 15 ∗ 2 , 15 ∗ 2 , 15 = 9 , 94 (\displaystyle 2.15*2.15*2.15=9.94).

9. Write down your answer. The result, written above the root sign, is the answer accurate to two decimal places. In our example, the cube root of 10 is 2.15. Check your answer by cubeing it: 2.15^3 = 9.94, which is approximately 10. If you need more precision, continue with the calculation (as described above).

   Part 2

   Extracting the cube root using the estimation method

   1. Use number cubes to determine upper and lower limits. If you need to take the cube root of almost any number, find the cubes (of some numbers) that are close to the given number.

      • For example, you need to take the cube root of 600. Since 8 3 = 512 (\displaystyle 8^(3)=512) And 9 3 = 729 (\displaystyle 9^(3)=729), then the value of the cube root of 600 lies between 8 and 9. Therefore, use the numbers 512 and 729 as the upper and lower limits of the answer.

   2. Estimate the second number. You found the first number thanks to your knowledge of cubes of integers. Now turn the integer into decimal, adding to it (after the decimal point) a certain number from 0 to 9. It is necessary to find a decimal fraction, the cube of which will be close to, but less than the original number.

      • In our example, the number 600 is located between the numbers 512 and 729. For example, add the number 5 to the first number found (8). The number you get is 8.5.
   3. • In our example: 8 , 5 ∗ 8 , 5 ∗ 8 , 5 = 614 , 1. (\displaystyle 8.5*8.5*8.5=614.1.)

   4. Compare the cube of the resulting number with the original number. If the cube of the resulting number is larger than the original number, try estimating the smaller number. If the cube of the resulting number is much smaller than the original number, estimate big numbers until the cube of one of them exceeds the original number.

      • In our example: 8 , 5 3 (\displaystyle 8.5^(3))> 600. So evaluate the smaller number to 8.4. Cube this number and compare it with the original number: 8 , 4 ∗ 8 , 4 ∗ 8 , 4 = 592 , 7 (\displaystyle 8.4*8.4*8.4=592.7). This result is less than the original number. So the cube root of 600 is between 8.4 and 8.5.

   5. Estimate the following number to improve your answer accuracy. For each number you estimated last, add a number from 0 to 9 until you get the exact answer. In each evaluation round, you need to find the upper and lower limits between which the original number lies.

      • In our example: 8 , 4 3 = 592 , 7 (\displaystyle 8.4^(3)=592.7) And 8 , 5 3 = 614 , 1 (\displaystyle 8.5^(3)=614.1). The original number 600 is closer to 592 than to 614. Therefore, to the last number you estimated, assign a figure that is closer to 0 than to 9. For example, such a number is 4. Therefore, cube the number 8.44.

   6. If necessary, estimate a different number. Compare the cube of the resulting number with the original number. If the cube of the resulting number is larger than the original number, try estimating the smaller number. In short, you need to find two numbers whose cubes are slightly larger and slightly smaller than the original number.

      • In our example 8 , 44 ∗ 8 , 44 ∗ 8 , 44 = 601 , 2 (\displaystyle 8.44*8.44*8.44=601.2). This is slightly larger than the original number, so estimate another (smaller) number, such as 8.43: 8 , 43 ∗ 8 , 43 ∗ 8 , 43 = 599 , 07 (\displaystyle 8.43*8.43*8.43=599.07). Thus, the cube root of 600 lies between 8.43 and 8.44.

   7. Follow the described process until you get an answer that you are happy with. Estimate the next number, compare it with the original, then, if necessary, estimate another number, and so on. Please note that each additional digit after the decimal point increases the accuracy of the answer.

      • In our example, the cube of 8.43 is less than 1 less than the original number. If you need more precision, cube 8.434 and get: 8, 434 3 = 599, 93 (\displaystyle 8,434^(3)=599,93), that is, the result is less than 0.1 less than the original number.

How many angry words were spoken to him? Sometimes it seems that the cube root is incredibly different from the square root. Actually the difference is not that big. Especially if you understand that they are only special cases common root nth degree.

However, problems may arise with its extraction. But most often they are associated with the cumbersomeness of calculations.

What do you need to know about the root of an arbitrary degree?

First, a definition of this concept. The nth root of some “a” is a number that, when raised to the power n, gives the original “a”.

Moreover, there are even and odd degrees at the roots. If n is even, then the radical expression can only be zero or a positive number. Otherwise there will be no real answer.

When the degree is odd, then there is a solution for any value of “a”. It may well be negative.

Secondly, the root function can always be written as a power, the exponent of which is a fraction. Sometimes this can be very convenient.

For example, “a” to the power 1/n will be the nth root of “a”. In this case, the base of the degree is always greater than zero.

Similarly, “a” to the power n/m will be represented as the mth root of “a n”.

Thirdly, all operations with powers are valid for them.

• They can be multiplied. Then the exponents add up.
• The roots can be divided. Degrees will need to be subtracted.
• And raise it to a power. Then they should be multiplied. That is, the degree that was, to the one to which they are raised.

What are the similarities and differences between square and cube roots?

They are similar, like siblings, only their degrees are different. And the principle of their calculation is the same, the only difference is how many times the number must be multiplied by itself in order to obtain the radical expression.

And the significant difference was mentioned a little higher. But it wouldn’t hurt to repeat it. The square is only extracted from a non-negative number. While calculating the cube root of a negative value is not difficult.

Extracting cube root on a calculator

Everyone has done this for square roots at least once. But what if the degree is “3”?

On a regular calculator there is only a button for square, but not for cubic. A simple search of numbers that are multiplied by themselves three times will help here. Did you get a radical expression? So this is the answer. Didn't work out? Pick again.

What is the engineering form of a calculator on a computer? Hooray, there's a cube root here. You can simply press this button, and the program will give you the answer. But that's not all. Here you can calculate not only the 2nd and 3rd degree roots, but also any arbitrary one. Because there is a button that has “y” in the root degree. That is, after pressing this key, you will need to enter another number, which will be equal to the degree of the root, and only then “=”.

Extracting cube roots manually

This method will be required when a calculator is not at hand or cannot be used. Then, in order to calculate the cube root of a number, you will need to make an effort.

First, see if a full cube is obtained from some integer value. Maybe the root is 2, 3, 5 or 10 to the third power?

1. Mentally divide the radical expression into groups of three digits from the decimal point. Most often you need a fractional part. If it is not there, then zeros must be added.
2. Determine the number whose cube is less than the integer part of the radical expression. Write it down in the intermediate answer above the root sign. And under this group place its cube.
3. Perform subtraction.
4. Add the first group of digits after the decimal point to the remainder.
5. In the draft, write down the expression: a 2 * 300 * x + a * 30 * x 2 + x 3. Here “a” is an intermediate answer, “x” is a number that is less than the resulting remainder with the numbers assigned to it.
6. The number “x” must be written after the decimal point of the intermediate answer. And write the value of this entire expression under the remainder being compared.
7. If the accuracy is sufficient, then stop the calculations. Otherwise, you need to return to point number 3.

An illustrative example of calculating a cube root

It is needed because the description may seem complicated. The figure below shows how to take the cube root of 15 to the nearest hundredth.

The only difficulty this method has is that with each step the numbers increase many times and counting in a column becomes more and more difficult.

1. 15> 2 3, means under whole part written 8, and above the root 2.
2. After subtracting eight from 15, the remainder is 7. Three zeros must be added to it.
3. a = 2. Therefore: 2 2 * 300 * x +2 * 30 * x 2 + x 3< 7000, или 1200 х + 60 х 2 + х 3 < 7000.
4. Using the selection method, it turns out that x = 4. 1200 * 4 + 60 * 16 + 64 = 5824.
5. Subtraction gives 1176, and the number 4 appears above the root.
6. Add three zeros to the remainder.
7. a = 24. Then 172800 x + 720 x 2 + x 3< 1176000.
8. x = 6. Evaluating the expression gives the result 1062936. Remainder: 113064, above root 6.
9. Add zeros again.
10. a = 246. The inequality turns out like this: 18154800x + 7380x 2 + x 3< 113064000.
11. x = 6. Calculations give the number: 109194696, Remainder: 3869304. Above root 6.

The answer is the number: 2.466. Since the answer must be given to the nearest hundredth, it must be rounded: 2.47.

An unusual way to extract cube roots

It can be used when the answer is an integer. Then the cube root is extracted by decomposing the radical expression into odd terms. Moreover, there should be the minimum possible number of such terms.

For example, 8 is represented by the sum of 3 and 5. And 64 = 13 + 15 + 17 + 19.

The answer will be a number that is equal to the number of terms. So the cube root of 8 will be equal to two, and of 64 - four.

If the root is 1000, then its decomposition into terms will be 91 + 109 + 93 + 107 + 95 + 105 + 97 + 103 + 99 + 101. There are 10 terms in total. This is the answer.

Engineering calculator online

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Undoubtedly, Web20calc will be of interest to that group of people who are looking for simple solutions dials in search engines request: mathematical online calculator. A free web application will help you instantly calculate the result of some mathematical expression, for example, subtract, add, divide, extract the root, raise to a power, etc.

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Extracting roots (square, cubic, and nth root);
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We have already sorted out a large number without a calculator. In this article we will look at how to extract the cube root (third degree root). Let me make a reservation that we are talking about natural numbers. How long do you think it takes to verbally calculate roots such as:

Quite a bit, and if you practice two or three times for 20 minutes, then you can extract any such root orally in 5 seconds.

*It should be noted that we are talking about numbers under the root that are the result of cubed natural numbers from 0 to 100.

We know that:

So, the number a that we will find is natural number from 0 to 100. Look at the table of cubes of these numbers (results of raising to the third power):

You can easily extract the cube root of any number in this table. What do you need to know?

1. These are cubes of numbers that are multiples of ten:

I would even say that these are “beautiful” numbers, they are easy to remember. It's easy to learn.

2. This is a property of numbers when they are producted.

Its essence lies in the fact that when raising a certain number to the third power, the result will have a peculiarity. Which one?

For example, let's cube 1, 11, 21, 31, 41, etc. You can look at the table.

1 3 = 1, 11 3 = 1331, 21 3 = 9261, 31 3 = 26791, 41 3 = 68921 …

That is, when we cube a number with a unit at the end, the result will always be a number with a unit at the end.

When you cube a number with a two at the end, the result will always be a number with an eight at the end.

Let's show the correspondence in the table for all numbers:

Knowledge of the two points presented is quite enough.

Let's look at examples:

Take the cube root of 21952.

This number is in the range from 8000 to 27000. This means that the result of the root lies in the range from 20 to 30. The number 29952 ends in 2. This option is only possible when a number with an eight at the end is cubed. Thus, the result of the root is 28.

Find the cube root of 54852.

This number is in the range from 27000 to 64000. This means that the result of the root lies in the range from 30 to 40. The number 54852 ends in 2. This option is only possible when a number with an eight at the end is cubed. Thus, the result of the root is 38.

Take the cube root of 571787.

This number is in the range from 512000 to 729000. This means that the result of the root lies in the range from 80 to 90. The number 571787 ends in 7. This option is only possible when a number with a three at the end is cubed. Thus, the result of the root is 83.

Take the cube root of 614125.

This number is in the range from 512000 to 729000. This means that the result of the root lies in the range from 80 to 90. The number 614125 ends in 5. This option is only possible when a number with a five at the end is cubed. Thus, the result of the root is 85.

I think that you can now easily extract the cube root of the number 681472.

Of course, extracting such roots orally takes a little practice. But by restoring the two indicated tablets on paper, you can easily extract such a root within a minute, in any case.

After you have found the result, be sure to check it (raise it to the third power). *Nobody canceled multiplication by column 😉

Actually Unified State Exam problems with such “ugly” roots, no. For example, you need to extract the cube root of 1728. I think this is no longer a problem for you.

If you know any interesting methods of calculations without a calculator, send them, I will publish them in due course.That's all. Good luck to you!

Sincerely, Alexander Krutitskikh.

P.S: I would be grateful if you tell me about the site on social networks.