Content

• Steps
• Part 1 of 3: Extracting the cube root with a simple example
• Part 2 of 3: Cube Root Estimation
• Part 3 of 3: Explaining the Calculation Process Described
• Tips
• Warnings
• What do you need

If you have a calculator at hand, you can easily extract the cube root of any number. But if you don't have a calculator, or you just want to impress others, manually extract the cube root. For most people, the process described here will seem rather complicated, but with practice it will become much easier to extract cube roots. Before you start reading this article, remember the basic mathematical operations and calculations with numbers in a cube.

Steps

Part 1 of 3: Extracting the cube root with a simple example

1. 1 Write down the task. Manual cube root extraction 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 extract the cube root. Divide the number into groups of three digits, and start counting with a decimal point. For example, you need to extract the cube root of 10. Write the number like this: 10,000,000. Additional zeros are used to improve the precision of the result.
   • Draw a root sign next to and above the number. Imagine these are the horizontal and vertical lines that you draw in long division. The only difference is the shape of the two characters.
   • Place a decimal point above the horizontal line. Do this directly above the decimal point of the original number.
2. 2 Remember the results of cubing integers. They will be used in calculations.
   • ${ displaystyle 1 ^ {3} = 1 * 1 * 1 = 1}$
   • ${ displaystyle 2 ^ {3} = 2 * 2 * 2 = 8}$
   • ${ displaystyle 3 ^ {3} = 3 * 3 * 3 = 27}$
   • ${ displaystyle 4 ^ {3} = 4 * 4 * 4 = 64}$
   • ${ displaystyle 5 ^ {3} = 5 * 5 * 5 = 125}$
   • ${ displaystyle 6 ^ {3} = 6 * 6 * 6 = 216}$
   • ${ displaystyle 7 ^ {3} = 7 * 7 * 7 = 343}$
   • ${ displaystyle 8 ^ {3} = 8 * 8 * 8 = 512}$
   • ${ displaystyle 9 ^ {3} = 9 * 9 * 9 = 729}$
   • ${ displaystyle 10 ^ {3} = 10 * 10 * 10 = 1000}$
3. 3 Find the first digit of the answer. Select an integer cube that is closest to but smaller than the first group of three digits.
   • In our example, the first group of three digits is 10. Find the largest cube that is less than 10. That 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 ${ displaystyle 2 ^ {3}}$ = 8 under 10. Draw a line and subtract 8 from 10 (as in long division). The result is 2 (this is the first remainder).
   • Thus, you have found the first number of the answer. Consider if the given result is accurate enough. In most cases, this will be a very rough answer. Cub the result to find out how close it is to the original number. In our example: ${ displaystyle 2 ^ {3}}$ = 8, which is not very close to 10, so the calculations need to be continued.
4. 4 Find the next digit of the answer. Add the second group of three numbers 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, the second group of three digits (000) must be added to the first remainder (2) to get the number 2000.
   • To the left of the vertical line, you write three numbers, the sum of which is equal to some first factor. Leave blank spaces for these numbers and put plus signs between them.
5. 5 Find the first term (out of three). In the first empty space, write down the result of multiplying 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 1200 (plus two more numbers to find).
6. 6 Find the second digit of the answer. Find out what number you need to multiply 1200 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 (second digit of the answer) after 2 and decimal comma above the root sign.
7. 7 Find the second and third terms (out of three). The factor 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. 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.
8. 8 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).
9. 9 Consider if the answer you received is accurate enough. Do this every time after completing the next 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 the answer, cub 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.
10. 10 Find the second factor. To practice your calculations and get a more accurate result, repeat the steps above.
    • Add the third group of three digits (000) to the second remainder (739). You will get the number 739000.
    • Multiply 300 by the square of the number written above the root sign (21): ${ displaystyle 300 * 21 ^ {2}}$ = 132300.
    • Find the third digit of the answer. Find out what number you need to multiply 132300 so that the result is close, but does not exceed 739000. That number is 5: 5 * 132200 = 661500. Write 5 (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: ${ displaystyle 3 * 21 * 5 * 10 = 3150}$.
    • Finally, square the last digit of your answer. In our example, the last digit of the answer is 5, so ${ displaystyle 5 ^ {2} = 25.}$
    • Thus, the second factor is: 132300 + 3150 + 25 = 135,475.
11. 11 Multiply the last digit of your answer by the second factor. After you have found the second factor and the third digit of the answer, proceed as follows:
    • Multiply the last digit of the answer by the factor found: 135475 * 5 = 677375.
    • Subtract: 739000 - 677375 = 61625.
    • Consider if the answer you received is accurate enough. To do this, cube it: ${ displaystyle 2.15 * 2.15 * 2.15 = 9.94}$.
12. 12 Write down your answer. The result written above the root sign is the answer with two decimal places. In our example, the cube root of 10 is 2.15. Check your answer by cubing it: 2.15 ^ 3 = 9.94, which is approximately 10. If you need more precision, continue the calculation (as described above).

Part 2 of 3: Cube Root Estimation

1. 1 Use cubes of numbers to determine the upper and lower limits. If you need to extract the cube root of almost any number, find cubes (of some numbers) that are close to the given number.
   • For example, you need to extract the cube root of 600. Since ${ displaystyle 8 ^ {3} = 512}$ and ${ displaystyle 9 ^ {3} = 729}$, then the cube root of 600 is between 8 and 9. Therefore, use 512 and 729 as the upper and lower limits of your answer.
2. 2 Estimate the second number. You found the first number thanks to your knowledge of the cubes of integers. Now convert an integer into a decimal fraction by assigning to it (after the decimal point) some digit from 0 to 9. You need 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 512 and 729. For example, to the first found number (8), add the number 5. You get the number 8.5.
3. 3 Estimate the resulting number by building it into a cube. Do this to check that the cube is close but not larger than the original number.
   • In our example: ${ displaystyle 8.5 * 8.5 * 8.5 = 614.1.}$
4. 4 Evaluate a different number if needed. 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 evaluating a lower number. If the cube of the resulting number is much smaller than the original number, evaluate the large numbers until the cube of one of them exceeds the original number.
   • In our example: ${ displaystyle 8.5 ^ {3}}$ > 600. Thus, estimate the smaller number 8.4. Cube this number and compare it with the original number: ${ displaystyle 8.4 * 8.4 * 8.4 = 592.7}$... This result is less than the original number. Thus, the cube root of 600 is between 8.4 and 8.5.
5. 5 Evaluate the next number to improve the accuracy of your answer. For each number that you rated 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 is.
   • In our example: ${ displaystyle 8.4 ^ {3} = 592.7}$ and ${ displaystyle 8.5 ^ {3} = 614.1}$... The original number 600 is closer to 592 than to 614. Therefore, to the last number that you estimated, add a digit that is closer to 0 than to 9. For example, this number is 4. Therefore, cub the number 8.44.
6. 6 Evaluate a different number if needed. 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 evaluating a lower number. In short, you need to find two numbers whose cubes are slightly larger and slightly smaller than the original number.
   • In our example ${ displaystyle 8.44 * 8.44 * 8.44 = 601.2}$... This is slightly larger than the original number, so evaluate another (smaller) number, for example 8.43: ${ displaystyle 8.43 * 8.43 * 8.43 = 599.07}$... Thus, the cube root of 600 is between 8.43 and 8.44.
7. 7 Follow this process until you get an answer that is satisfactory to you. Evaluate the next number, compare it with the original, then evaluate another number if necessary, and so on. Note that each additional digit after the decimal point increases the precision of your answer.
   • In our example, the cube of the number 8.43 is less than the original number by less than 1. If you need more precision, cube the number 8.434 and get that ${ displaystyle 8,434 ^ {3} = 599.93}$, that is, the result is less than 0.1 less than the original number.

Part 3 of 3: Explaining the Calculation Process Described

1. 1 Remember the binomial series. The binomial series is the result of raising a binomial (binomial) to a certain power, in this case to a cube. To understand the cube root extraction algorithm described here, first remember how a binomial is cube. Chances are, you learned this in school (and probably soon forgot, as most people do). Variables ${ displaystyle A}$ and ${ displaystyle B}$ mark some single digits. Then the two-digit number can be written as a binomial ${ displaystyle (10A + B)}$.
   • Here the member ${ displaystyle 10A}$ represents the tens place, that is, if ${ displaystyle A}$ Is any single-digit number, then ${ displaystyle 10A}$ - this is already the corresponding two-digit number. For example, if ${ displaystyle A}$ = 2, and ${ displaystyle B}$ = 6, then ${ displaystyle (10A + B)}$ = 26, that is, you got a two-digit number 26.
2. 2 Cube the binomial. Do this in order to understand the cube root extraction process described in the first section. Calculate ${ displaystyle (10A + B) ^ {3}}$ = ${ displaystyle (10A + B) * (10A + B) * (10A + B)}$ = ${ displaystyle 1000A ^ {3} + 300A ^ {2} B + 30AB ^ {2} + B ^ {3}}$ (here we have omitted several stages of cube construction, so as not to clutter up the article with calculations).
   • A detailed explanation can be found here.
3. 3 Understand the long division algorithm. Note that the cube root method described here is very similar to long division. When dividing in a column, you need to find the number (quotient), when multiplied by the divisor, you get the dividend. In the described method, the result of extracting the cube root (it is written above the root sign) acts as a quotient. That is, the result of extracting the cube root can be represented as a binomial (10A + B). The exact values ​​of A and B are not important at this stage: just remember that the result can be written as a binomial.
4. 4 Look at the binomial range. It is the sum of four monomials, thanks to which you can understand the principle of operation of the cube root extraction algorithm. Please note that the multiplier for each step of extracting the root is equal to the sum of the four terms that need to be calculated and added.
   • The factor for the first term is 1000. To calculate the first digit of the answer, you first find the cube of an integer that is closest to but less than a certain number (namely the first group of three digits). This defines the 1000A ^ 3 member of the binomial series.
   • The multiplier of the second term of the binomial series is the number 300 (${ displaystyle 3 * 10 ^ {2}}$ = 300). Recall that at each stage of cube root extraction, the corresponding digit (s) of the answer were multiplied by 300.
   • The second term at each stage of root extraction is determined by the third term of the binomial series, which is equal to 30AB ^ 2.
   • The third term at each stage of root extraction is determined by the fourth term of the binomial series, which is equal to B ^ 3.
5. 5 Note the increase in the accuracy of the answer. The more stages of root extraction you go through, the more accurate the answer will be. For example, in this article, you needed to extract the cube root of 10. At the first stage, the answer is 2, since ${ displaystyle 2 ^ {3}}$ = 8, which is close, but less than 10. At the second stage, the answer is 2.1, because ${ displaystyle 2.1 ^ {3} = 9.261}$, which is much closer to 10. At the third stage, the answer is 2.15, since ${ displaystyle 2.15 ^ {3} = 9.94}$... You can continue the calculation using groups of three digits to improve the accuracy of your answer.

Tips

• Practice to master the methods described. The more you practice, the faster you will get through the calculations.

Warnings

• It is quite easy to make a mistake in the computation process. So be sure to check the answer.

What do you need

• Pen or pencil
• Paper
• Ruler
• Eraser