Content

• To step
• Part 1 of 3: Working out an example assignment
• Part 2 of 3: Finding the cube root through repeated estimating
• Tips
• Warnings
• Necessities

Using a calculator, calculating the cube root of any number is no more than pressing a few keys. But maybe you don't have a calculator or want to impress your friends with your ability to work out a cube root freehand. There is a method that looks a bit tough at first glance, but works very simply with a little practice. It is useful to have some ready knowledge in the field of arithmetic skills and calculating cubic numbers.

To step

Part 1 of 3: Working out an example assignment

1. Draw up the problem. Solving the cube root of a number will look like solving a long division, with some differences here and there. The first step is to write down the statement correctly.
   • Write down the number for which you want to determine the cube root of. Write the numbers in groups of three, with the comma being the starting point. In this example, you are going to determine the cube root of 10. Write this as 10.000000. The zeros are needed for the accuracy of the answer.
   • Draw a cube square root over the number. This serves the same purpose as the line in long division. The only difference is the shape of the symbol.
   • Place a comma above the line, directly above the comma in the original number.
2. Know the cubes of the units. You are going to use these in your calculations. It concerns the following third powers:
   • ${ displaystyle 1 ^ {3} = 1 * 1 * 1 = 1}$Determine the first digit of your answer. Select a number that, to the cube, gives the greatest possible result that is less than the first set of three numbers.
     • In this example, the first set of three numbers multiplied together is equal to 10. Find the greatest cube that is less than 10. That is 8, and its cube root is 2.
     • Write the number 2 above the square root sign, above the number 10. Write down the value of ${ displaystyle 2 ^ {3}}$Make the setup for the next digit. Write the next group of three numbers in the rest, and draw a short vertical line to the left of the resulting number. This will be the number we use to determine the next digit in the solution of your cube root. In this example, this becomes 2000, which is created from the remainder 2 of the previous subtraction sum, with the group of three zeros you took down.
       • To the left of the vertical line, write the solution of the next divisor, as the sum of three separate numbers. Indicate the empty spaces for these numbers, by underlining three blank spots with plus signs underneath.
     • Find the start of the next divisor. For the first part of the divisor, write three hundred times the square of whatever is above the square root sign. In this case it is 2; 2 ^ 2 is 4, and 4 * 300 = 1200. So write your 1200 in the first blank space. The divisor for this step of the solution becomes 1200, plus something else that you will calculate in a moment.
     • Find the next number in your cube root. Find the next digit of your solution by selecting what you can multiply by the divisor (1200's and something else), and then subtract it from the remainder of 2000. This can only be 1, because 2 times 1200 equals 2400, which is greater than 2000. Write the number 1 in the next space above the square root sign.
     • Find the remainder of the divisor. The divisor in this step of the solution consists of three parts. The first part is the 1200 you already have. You will now need to add two more terms to complete the divisor.
       • Now calculate 3 times 10 times each of the two digits in your solution above the square root sign. For this simple exercise, that means 3 * 10 * 2 * 1, which is equal to 60. Add this to the 1200 you already had and you get 1260.
       • Finally, add the square of the last digit. In this example it is 1; and 1 ^ 2 is still 1. So the total divisor is 1200 + 60 + 1, or 1261. Write this to the left of the vertical line.
     • Multiply and subtract. Round off this part of the solution by multiplying the last digit of your solution - in this case, the number 1 - times the divisor you just calculated (1261). 1 * 1261 = 1261. Write this below 2000 and subtract 1261 to get 739.
     • Decide to go further for a more accurate answer. After completing the subtraction of each step, you need to check if your answer is exact enough. For the cube root of 10, after the first minus sum, the cube root was only 2, which isn't really exact. Now, after the second round, the solution is 2.1.
       • You can check the precision of this result using the cube: 2.1 * 2.1 * 2.1. The result is 9.261.
       • If you think the result is exact enough, you can stop. If you want a more precise answer, you have to go through another round.
     • Determine the divisor for the next round. In this case, for more practice and a more precise answer, repeat the steps for another round, as follows:
       • Bring down the next group of three numbers. In this case, these are three zeros, which come after the remainder 739 to form 739,000.
       • Begin the divisor with 300 times the square of the number currently above the square root sign. This is ${ displaystyle 300 * 21 ^ {2}}$Multiply the divisor by the result. After calculating the divisor in this next round and expanding your solution with one more digit, proceed as follows:
         • Multiply the divisor by the last digit of your solution. 135,475 * 5 = 677,375.
         • Subtract. 739,000-677,375 = 61,625.
         • Consider whether the solution 2.15 is exact enough. Calculate the cube of it and you will get ${ displaystyle 2.15 * 2.15 * 2.15 = 9.94}$Write down your final answer. The result above the square root is the cube root, to an accuracy of three significant digits. In this example, the cube root of 10 equals 2.15. Check this by calculating 2.15 ^ 3 = 9.94 which can be rounded up to 10. If you need a more accurate answer, keep doing this until you are satisfied.

Part 2 of 3: Finding the cube root through repeated estimating

1. Use cubic numbers to set the upper and lower limits. When asked for the cube root of a given number, start by choosing a cube that is as close to it as possible, without being greater than your target number.
   • For example, if you want to find the cube root of 600, remember (or use a cube cube) that ${ displaystyle 8 ^ {3} = 512}$Estimate the next digit. You delete the first digit through your knowledge of certain cubic numbers. For the next digit, estimate a number between 0 and 9 based on where your target number falls between the two limit numbers.
     • In the example problem, 600 (your target number) falls about halfway between the limit numbers 512 and 729. So you choose 5 as your next number.
   • Test your estimate by determining the cube of it. Try multiplying the estimate you are currently working with to find out how close you are to the target number.
     • In this example, you are multiplying ${ displaystyle 8.5 * 8.5 * 8.5 = 614.1.}$Adjust your estimate as necessary. After raising to the cube of your latest guess, check the result against your target number. If the result is greater than the target, your estimate should be smaller. If the result is less than the goal, you have to adjust it upwards until you reach the goal.
       • For example, in this statement ${ displaystyle 8.5 ^ {3}}$Estimate the next digit for a more accurate answer. Continue this procedure of estimating numbers from 0 to 9 until your answer is as accurate as you want. Before each round of estimation, you start by checking the position of your last calculation between the boundary numbers.
         • In this example exercise, your last round of calculations shows that ${ displaystyle 8.4 ^ {3} = 592.7}$Continue estimating and adjusting. Do this as many times as needed, raise your guess to cubic power and see how it compares to the target number. Look for numbers that are just below or just above the target number.
           • For this example exercise, you start by noting that ${ displaystyle 8.44 * 8.44 * 8.44 = 601.2}$Continue until you reach the desired accuracy. Continue estimating, comparing and re-estimating for as long as necessary until your solution is as accurate as you want. Note that with each decimal, your target numbers get closer and closer to the actual number.
             • For the cube root of 600 example, assuming two decimal numbers, you are less than 1 away from the target number by 8.43. If you continue to three decimal places, you will see that ${ displaystyle 8.434 ^ {3} = 599.93}$Review Newton's binomium. To understand why this algorithm works for determining cube roots, you must first think back to what cube looks like as binomial. You probably learned this in high school math (and like most people, you probably quickly forgot about this). Select two variables ${ displaystyle A}$Write the binomial in cubic form. We are now working backwards by first determining the cube and then looking at why the cube root solution works. We need the values ​​of ${ displaystyle (10A + B) ^ {3}}$Know the meaning of long division. Note that the cube root method works just like long division. In long division you see that two factors multiplied together give the number you started with. In this calculation, the number you are looking for (the number that eventually appears above the square root) is the cube root. That means it equals the term (10A + B). The actual A and B are now irrelevant, as long as you understand the relationship with the answer.
             • View the extended version. When you look at Newton's binomium, you can see why the cube root algorithm is correct. See how the divisor at each step of the algorithm equals the sum of the four terms you need to calculate and add. These terms arise as follows:
               • The first term contains a multiple of 1000. You first choose a number that could be raised to the cube and still remain within the range of the long division as the first number. This gives the term 1000A ^ 3 in the binomial.
               • The second term of Newton's binomium has 300 as its coefficient. (This comes from ${ displaystyle 3 * 10 ^ {2}}$Watch accuracy grow. When working out long division, each step you complete gives great accuracy to your answer. For example, the example problem worked in this article is for determining the cube root of 10. In the first step, the solution is 2, because ${ displaystyle 2 ^ {3}}$ comes close, but is less than 10. In fact, it holds ${ displaystyle 2 ^ {3} = 8}$. After the second round, your solution is 2.1. Once you have worked this out, you will get ${ displaystyle 2.1 ^ {3} = 9,261}$, which is much closer to the desired result (10). After the third round, you have 2.15, which gives you ${ displaystyle 2.15 ^ {3} = 9.94}$. Keep working in groups of three numbers and you will get as accurate an answer as you want.

Tips

• Like anything else, your math skills will improve with practice. The more you practice, the better you will be able to do these kinds of calculations.

Warnings

• It's easy to make a mistake with this. Check your work carefully and go through the elaboration again.

Necessities

• Pen or pencil
• Paper
• Ruler
• Eraser