Write down the number whose cube root you want to find. Write the digits in groups of three, using the decimal point as your starting place. For this example, you will find the cube root of 10. Write this as 10. 000 000. The extra 0s are to allow precision in the solution. Draw a cube root radical sign over the number. This serves the same purpose as the long division bar line. The only difference is the shape of the symbol. Place a decimal point above the bar line, directly above the decimal point in the original number.
13=1∗1∗1=1{\displaystyle 1^{3}=111=1} 23=2∗2∗2=8{\displaystyle 2^{3}=222=8} 33=3∗3∗3=27{\displaystyle 3^{3}=333=27} 43=4∗4∗4=64{\displaystyle 4^{3}=444=64} 53=5∗5∗5=125{\displaystyle 5^{3}=555=125} 63=6∗6∗6=216{\displaystyle 6^{3}=666=216} 73=7∗7∗7=343{\displaystyle 7^{3}=777=343} 83=8∗8∗8=512{\displaystyle 8^{3}=888=512} 93=9∗9∗9=729{\displaystyle 9^{3}=999=729} 103=10∗10∗10=1000{\displaystyle 10^{3}=101010=1000}
In this example, the first set of three numbers is 10. Find the largest perfect cube that is less than 10. That number is 8, and its cube root is 2. Write the number 2 above the radical bar line, over the number 10. Write the value of 23{\displaystyle 2^{3}}, which is 8, underneath the number 10, draw a line and subtract, just as you would in long division. The result is a 2. After the subtraction, you have the first digit of your solution. You need to decide if this one digit is a precise enough result. In most cases, it will not be. You can check by cubing the single digit and decide if that is close enough to the result you wanted. Here, because 23{\displaystyle 2^{3}} is only 8, not very close to 10, you should continue.
To the left of the vertical line, you will be solving the next divisor, as the sum of three separate numbers. Draw the spaces for these numbers by making three blank underlines, with plus symbols between them.
Now calculate 3 times 10 times each of the two digits that are in your solution above the radical sign. For this sample problem, that means 3102*1, which is 60. Add this to the 1200 that you already have to make 1260. Finally, add the square of the last digit. For this example, that is a 1, and 1^2 is still 1. The total divisor is, therefore 1200+60+1, or 1261. Write this to the left of the vertical line.
You can check the precision of this result by cubing 2. 12. 12. 1. The result is 9. 261. If you believe your result is precise enough, you can quit. If you want a more precise answer, then you need to proceed with another round.
Drop down the next group of three digits. In this case, these are three 0s, which will follow the 739 remainder to give 739,000. Begin the divisor with 300 times the square of the number currently above the radical line. This is 300∗212{\displaystyle 30021^{2}}, which is 132,300. Select the next digit of your solution so that you can multiply it by 132,300 and have less than the 739,000 of your remainder. A good choice would be 5, since 5132,300=661,500. Write the digit 5 in the next space above the radical line. Find 3 times the prior number above the radical line, 21, times the last digit you just wrote, 5, times 10. This gives 3∗21∗5∗10=3,150{\displaystyle 3215*10=3,150}. Finally, square the last digit. This is 52=25. {\displaystyle 5^{2}=25. } Add the parts of your divisor to get 132,300+3,150+25=135,475.
Multiply the divisor by the last digit of your solution. 1354755=677,375. Subtract. 739,000-677,375=61,625. Consider whether the solution of 2. 15 is precise enough. Cube it to get 2. 15∗2. 15∗2. 15=9. 94{\displaystyle 2. 152. 15*2. 15=9. 94}.
For example, if you want to find the cube root of 600, recall (or use a table of cube numbers) that 83=512{\displaystyle 8^{3}=512} and 93=729{\displaystyle 9^{3}=729}. Therefore, the solution for the cube root of 600 must be something between 8 and 9. You will use the numbers 512 and 729 as upper and lower boundaries for your solution.
In the working example, the target of 600 falls about halfway between the boundary numbers of 512 and 729. So, select 5 for your next digit.
In this example, multiply 8. 5∗8. 5∗8. 5=614. 1. {\displaystyle 8. 58. 58. 5=614. 1. }
For example, in this problem, 8. 53{\displaystyle 8. 5^{3}} is greater than the target of 600. So you should reduce the estimate to 8. 4. Cube this number and compare to your target. You will find that 8. 4∗8. 4∗8. 4=592. 7{\displaystyle 8. 48. 48. 4=592. 7}. This is now lower than your target. Therefore, you know that the cube root of 600 must be at least 8. 4 but less than 8. 5.
In this working example, your last round of calculations shows that 8. 43=592. 7{\displaystyle 8. 4^{3}=592. 7}, while 8. 53=614. 1{\displaystyle 8. 5^{3}=614. 1}. The target of 600 is slightly closer to 592 than it is to 614. So for your next guess, begin by choosing a number slightly less than halfway between 0 and 9. A good guess would be 4, for a cube root estimate of 8. 44.
For this working example, begin by finding that 8. 44∗8. 44∗8. 44=601. 2{\displaystyle 8. 448. 448. 44=601. 2}. This is just barely above the target, so drop down and test 8. 43. This will give you 8. 43∗8. 43∗8. 43=599. 07{\displaystyle 8. 438. 438. 43=599. 07}. Therefore, you know that the cube root of 600 is something more than 8. 43 and less than 8. 44.
For the example of the cube root of 600, when you used two decimal places, 8. 43, you were away from the target by less than 1. If you continue to a third decimal place, you would find that 8. 4343=599. 93{\displaystyle 8. 434^{3}=599. 93}, less than 0. 1 from the true answer.
Using the term 10A{\displaystyle 10A} is what creates a two digit number. Whatever digit you select for A{\displaystyle A}, 10A{\displaystyle 10A} will put that digit into the tens column. For example, if A{\displaystyle A} is 2 and B{\displaystyle B} is 6, then (10A+B){\displaystyle (10A+B)} becomes 26. [10] X Research source
For more about expanding the binomial to get this result, you can see Multiply Binomials. For a more advanced, shortcut version, read Calculate (x+y)^n with Pascal’s Triangle.
The first term contains a multiple of 1000. You first a number that could be cubed and stay within the range for the long division for the first digit. This provides the term 1000A^3 in the binomial expansion. The second term of the binomial expansion has the coefficient of 300. (This actually comes from 3∗102{\displaystyle 3*10^{2}}. ) Recall that in the cube root calculation, the first digit in each step is multiplied by 300. The second digit in each step of the cube root calculation comes from the third term of the binomial expansion. In the binomial expansion, you can see the term 30AB^2. The final digit of each step is the term B^3.
