If you have a calculator at hand, extracting the cube root of any number will not be a problem. But if you don't have a calculator, or if you just want to impress others, you can do the cube root by hand. To most people, the process described here will seem rather 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 numbers in the cube.
Steps
Part 1
Extracting the cube root simple example- Write down the number from which you want to take the cube root. Divide the number into groups of three digits, and start counting with a decimal point. For example, you need to take the cube root of 10. Write this number like this: 10, 000,000. Additional zeros designed to improve the accuracy of the result.
- Near and above the number, draw the sign of the root. Think of it as the horizontal and vertical lines you draw when you divide into a column. 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.
-
Remember the results of cuing 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)
-
Find the first digit of the answer. Choose the integer cube 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 in normal long division). The result is 2 (this is the first remainder).
- Thus, you have found the first digit of the answer. Consider whether this result is accurate enough. 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.
-
Find the next digit in the answer. Attribute the second group of three digits to the first remainder, and draw a vertical line to the left of the resulting number. With the help of the received number you will find the second digit of the answer. In our example, the first remainder (2) needs to be assigned a second group of three digits (000) to get the number 2000.
- To the left of the vertical line, you will write three numbers whose sum is equal to some first factor. Leave blank spaces for these numbers, and put plus signs between them.
-
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 empty space. The first term is 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 to, but not more than 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 over the sign of the root.
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 for 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 you 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. So the first factor is the sum of the following numbers: 1200 + 60 + 1 = 1261. Write this number to the left of the vertical bar.
-
Multiply and subtract. Multiply the last digit of the answer (in our example it is 1) by the factor found (1261): 1 * 1261 = 1261. Write this number under 2000 and subtract it from 2000. You will get 739 (this is the second remainder).
-
Consider whether the answer you get is accurate enough. Do this every time after you complete another subtraction. After the first subtraction, the answer was 2, which is not an exact 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 calculation; otherwise, do another subtraction.
-
Find the second multiplier. 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 that is 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 not more than 739000. This number is 5: 5 * 132200 = 661500. Write 5 (the third digit of the answer) after 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.
-
Multiply the last digit of your answer by the second factor. After you have found the second multiplier and the third digit of the answer, proceed as follows:
- Multiply the last digit of the answer by the multiplier found: 135475*5 = 677375.
- Subtract: 739000-677375 = 61625.
- Consider whether the answer you get 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).
-
Write down the answer. The result written above the root sign is the answer to two decimal places. In our example, the cube root of 10 is 2.15. Check your answer by cubed: 2.15^3 = 9.94, which is approximately 10. If you need more precision, continue with the calculation (as described above).
Part 2
Extraction of the cube root by the method of estimates-
Use number cubes to determine the upper and lower limits. If you need to extract 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 cube root of 600 lies between 8 and 9. So use 512 and 729 as upper and lower limits for your answer.
-
Estimate the second number. You found the first number thanks to the knowledge of the cubes of integers. Now turn the integer into decimal, adding to it (after the decimal point) some figure from 0 to 9. It is necessary to find a decimal fraction, the cube of which will be close, but less than the original number.
- In our example, the number 600 is between the numbers 512 and 729. For example, add the number 5 to the first number found (8). You get the number 8.5.
-
- In our example: 8 , 5 ∗ 8 , 5 ∗ 8 , 5 = 614 , 1. (\displaystyle 8.5*8.5*8.5=614.1.)
-
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, evaluate 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 estimate the lower number 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. Thus, the value of the cube root of 600 lies between 8.4 and 8.5.
-
Estimate the next number to improve the accuracy of your answer. For each number you last estimated, add a number from 0 to 9 until you get an accurate 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 it is to 614. Therefore, to the last number you estimated, add a number that is closer to 0 than to 9. For example, this number is 4. So cube the number 8.44.
-
If necessary, evaluate another 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 evaluate 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 value of the cube root of 600 lies between 8.43 and 8.44.
-
Follow the process described until you get an answer that is accurate to your satisfaction. Evaluate the next number, compare it to the original, then evaluate another number if necessary, and so on. Note that each additional digit after the decimal point increases the accuracy of the answer.
- In our example, the cube of the number 8.43 is less than 1 less than the original number. If you need more precision, cube the number 8.434 and get that 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.
-
Write down the task. Extracting the cube root manually is similar to long division, but with some nuances. First, write down the task in a specific form.
The nth root of a number x is a non-negative number z that, when raised to the nth power, becomes x. The definition of the root is included in the list of basic arithmetic operations that we get acquainted 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 operation of extracting the root with the letter r with a horizontal bar above the radical expression. In the 16th century, the designation V was introduced, which gradually replaced the sign r, but the horizontal line was preserved. It is easy to type in a printing house or write by hand, but in electronic publishing and programming it has spread letter designation root - 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. Two in this case is the square root 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 and get 144, and the radical 144 will be 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, so when searching for answers in such problems, you need to specify both roots. When solving 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 not periodic, which causes some difficulties in calculating such radicals.
The school mathematics course states that you cannot take square roots from negative numbers. As we learn in the high school course of mathematical analysis, this can and should be done - this is what complex numbers are needed for. However, our program is designed to extract real values of the roots, so it does not calculate even-degree radicals from negative numbers.
cube root
The cubic radical of a number x is the 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 four times by itself and get 4 × 4 × 4 = 64. Obviously, four is the cube root of 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 the rest of the numbers, cube roots are irrational numbers. Unlike square radicals, cube roots, like any odd roots, can be taken from negative numbers. It's all about numbers less than zero. A minus by a minus gives a plus - a rule known from the school bench. A minus times a plus makes a minus. If you multiply negative numbers an odd number of times, then the result will also be negative, therefore, extract the odd radical from negative number nothing prevents us.
However, the calculator program works differently. In fact, extracting a root is raising to the inverse power. The square root is considered as raising to the power of 1/2, and the cube - 1/3. The formula for raising to the power of 1/3 can be reversed and expressed as 2/6. The result is the same, but it is impossible to extract such a root from a negative number. So our calculator calculates arithmetic roots only from positive numbers.
Nth root
Such an ornate way of calculating radicals allows you to determine the roots of any degree from any expression. You can extract the fifth root of the cube of a number, or the 19th radical of a number to the 12th. All this is elegantly implemented as exponentiation to the power of 3/5 or 12/19, respectively.
Consider an example
Square diagonal
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 square 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 "Power (n)" cell. As a result, we get the expression sqrt (2) = 1.4142. Thus, for a rough estimate of the diagonal of a square, it is enough to multiply its side by 1.4142.
Conclusion
The search for a radical is a standard arithmetic operation, without which scientific or design calculations are indispensable. Of course, we do not need to determine the roots to solve everyday problems, but our online calculator will definitely come in handy for schoolchildren or students to check their homework in algebra or calculus.
When solving some technical problems, it may be necessary to calculate the root third degrees. Sometimes this number is also called the cube root. root third degrees from a given number such a number is called, the cube (third degree) of which is equal to the given one. That is, if y is the root third degrees numbers x, then the following condition must be satisfied: y?=x (x is equal to y cube).
You will need
- calculator or computer
Instruction
- To calculate the root third degrees use the calculator. It is desirable that this is not an ordinary calculator, but a calculator used for engineering calculations. However, even on such a calculator you will not find a special button for extracting the root third degrees. So use a function to raise a number to a power. Extracting the root third degrees corresponds to raising to the power of 1/3 (one third).
- To raise a number to the power of 1/3, type the number itself on the calculator's keyboard. Then press the "exponentiation" key. Such a button, depending on the type of calculator, may look like xy (y - in the form of a superscript). Since most calculators do not have the ability to work with ordinary (non-decimal) fractions, instead of the number 1/3, type its approximate value: 0.33. To get greater accuracy of calculations, it is necessary to increase the number of "triples", for example, dial 0.33333333333333. Then, press the "=" button.
- To calculate the root third degrees on a computer, use the standard Windows calculator. The procedure is completely similar to that described in the previous paragraph of the instruction. The only difference is the designation of the exponentiation button. On a "computer" calculator, it looks like x ^ y.
- If root third degrees If you have to calculate systematically, then use MS Excel. To calculate the root third degrees in Excel, enter the “=” sign in any cell, and then select the “fx” icon - inserting a function. In the window that appears, in the "Select a function" list, select the line "DEGREE". Click the OK button. In the newly appeared window, enter in the line "Number" the value of the number from which you want to extract the root. In the line "Degree" enter the number "1/3" and click "OK". The desired value of the cube root from the original number will appear in the cell of the table.
How many angry words were uttered against him? Sometimes it seems that the cube root is incredibly different from the square. In fact, the difference is not that big. Especially if you understand that they are only special cases of a common root of the nth degree.
But with its extraction there can be problems. But most often they are associated with the cumbersomeness of calculations.
What you need to know about the root of an arbitrary degree?
First, the definition of this concept. The root of the nth degree of some "a" is a number that, when raised to the power of n, gives the original "a".
Moreover, there are even and odd degrees at the roots. If n is even, then the root expression can only be zero or positive number. Otherwise, there will be no real response.
When the degree is odd, then there is a solution for any value of "a". It may well be negative.
Secondly, the function of the root can always be written as a degree, the indicator of which is a fraction. Sometimes this is very convenient.
For example, "a" to the power of 1 / n will just be the n-th 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 m-th root of "a n".
Thirdly, all actions with powers are valid for them.
- They can be multiplied. Then the exponents add up.
- Roots can be divided. The degrees will need to be subtracted.
- And raise 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 degree is different. And the principle of their calculation is the same, the only difference is how many times the number must be multiplied by itself to get the root expression.
A significant difference was mentioned a little higher. But it won't hurt to repeat. The square is only extracted from a non-negative number. While calculating the cube root of a negative value is not difficult.
Extracting the cube root on a calculator
Everyone has done this for the square root at least once. But what if the degree is "3"?
On a conventional calculator, there is only a button for a square one, but no cubic one. A simple enumeration of numbers that are multiplied by themselves three times will help here. Got a root expression? So this is the answer. Did not work out? Pick up again.
And what about the engineering form of a calculator in a computer? Hooray, there is a cube root here. You can simply press this button, and the program will give you an answer. But that's not all. Here you can calculate the root not only of 2 and 3 degrees, but also of any arbitrary one. Because there is a button that has a “y” in the degree of the root. 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 “=”.
Manual cube root extraction
This method is required when a calculator is not at hand or cannot be used. Then, in order to calculate the cube root of the number, you will need to make an effort.
First, see if the full cube is obtained from some integer value. Maybe under the root is 2, 3, 5 or 10 to the third power?
- Mentally divide the root expression into groups of three digits from the decimal point. Most often, a fractional part is needed. If not, then add zeros.
- Determine the number whose cube is less than the integer part of the radical expression. Write it in an intermediate answer above the root sign. And under this group, place his cube.
- Perform subtraction.
- Attribute to the remainder the first group of digits after the decimal point.
- 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 numbers assigned to it.
- 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.
- If the accuracy is sufficient, then stop the calculations. Otherwise, you need to return to point number 3.
An illustrative example of calculating the cube root
It is needed because the description may seem complicated. The figure below shows how to extract the cube root of 15 with hundredths precision.
The only difficulty that this method has is that with each step the numbers increase many times over and it becomes more and more difficult to count in a column.
- 15> 2 3, so 8 is written under the integer part, and 2 above the root.
- After subtracting eight from 15, the remainder is 7. Three zeros must be attributed to it.
- a \u003d 2. Therefore: 2 2 * 300 * x + 2 * 30 * x 2 + x 3< 7000, или 1200 х + 60 х 2 + х 3 < 7000.
- The selection method turns out that x \u003d 4. 1200 * 4 + 60 * 16 + 64 \u003d 5824.
- Subtraction gives 1176, and the number 4 appeared above the root.
- Assign three zeros to the remainder.
- a \u003d 24. Then 172800 x + 720 x 2 + x 3< 1176000.
- x = 6. Evaluation of the expression gives the result 1062936. Remainder: 113064, over the root 6.
- Assign zeros again.
- a \u003d 246. The inequality turns out like this: 18154800x + 7380x 2 + x 3< 113064000.
- x \u003d 6. Calculations give the number: 109194696, Remainder: 3869304. Above the root 6.
The answer is a number: 2.466. Since the answer must be given to hundredths, it must be rounded: 2.47.
An unusual way to extract the cube root
It can be used when the answer is an integer. Then the cube root is extracted by expanding the radical expression into odd terms. Moreover, such terms should be the minimum possible number.
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 there is 1000 under the root, then its expansion into terms will be 91 + 109 + 93 + 107 + 95 + 105 + 97 + 103 + 99 + 101. There are 10 terms in total. This is the answer.
From a large number without a calculator, we have already sorted it out. In this article, we will look at how to extract the cube root (root of the third degree). Note that we are talking about natural numbers. How long do you think it takes to verbally calculate such roots as:
Quite a bit, and if you practice two or three times for 20 minutes, then you can extract any such root in 5 seconds orally.
*It should be noted that we are talking about such numbers under the root, which are the result of raising natural numbers from 0 to 100 into a cube.
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 (the 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 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 multiplied.
Its essence lies in the fact that when a certain number is raised to the third power, the result will have a singularity. What?
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, we will always end up with a number with a unit at the end.
When you cube a number ending in a 2, the result will always be a number ending in an 8.
Let's show the correspondence in the table for all numbers:
Knowing the two points presented is enough.
Consider examples:
Extract 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 with 2. This option is only possible when a number with an eight at the end is cubed. So the root result is 28.
Extract 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 with 2. This option is only possible when a number with an eight at the end is cubed. So the root result is 38.
Extract 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 with 7. This option is only possible when a number with a three at the end is cubed. So the root result is 83.
Extract 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 with 5. This option is only possible when a number with a five at the end is cubed. So the root result is 85.
I think that now you can easily extract the cube root of the number 681472.
Of course, extracting such roots orally takes a little practice. But having restored 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 degree). * Multiplication by a column has not been canceled 😉
At the USE itself, there are no problems with such “ugly” roots. For example, you need to extract the cube root of 1728. I think that this is not a problem for you now.
If you know some interesting calculation methods without a calculator, send it, I will publish it in due course.That's all. Good luck to you!
Sincerely, Alexander Krutitskikh.
P.S: I would be grateful if you tell about the site in social networks.
