How To Find Cube Root By Division Method

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With the use of calculators, finding the cube root of any number may be simply buttons away. But perhaps you don't take a calculator, or you lot want to print your friends with the ability to calculate a cube root by paw. There is a procedure that appears a flake laborious at first, but with practice it works adequately easily. It is helpful if you remember some basic math skills and some algebra virtually cube numbers.

1. ane

   Set upwards the trouble. Solving the cube root of a number is going to look like solving a long partition problem, with a few special differences. The starting time step is to fix the problem in the proper format.^[1]

   • Write downward the number whose cube root you desire to find. Write the digits in groups of three, using the decimal bespeak as your starting place. For this example, y'all will find the cube root of 10. Write this every bit 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.
   • Identify a decimal point in a higher place the bar line, directly above the decimal point in the original number.

2. 2

   Know the cubes of single digit numbers. You volition utilize these in the computations. These cubes are equally follows:

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3. three

   Notice the first digit of your solution. Select a number that, when cubed, gives the largest possible result less than the first gear up of three numbers.^[2]

   • In this example, the starting time gear up of three numbers is ten. Observe the largest perfect cube that is less than 10. That number is 8, and its cube root is 2.
   • Write the number 2 higher up the radical bar line, over the number 10. Write the value of ${\displaystyle 2^{3}}$ , which is 8, underneath the number ten, describe a line and subtract, just equally yous would in long sectionalisation. The result is a 2.
   • Afterward the subtraction, yous have the kickoff digit of your solution. You need to decide if this one digit is a precise enough result. In most cases, it will non be. You lot tin can bank check past cubing the unmarried digit and decide if that is close enough to the result you wanted. Here, because ${\displaystyle 2^{three}}$ is only 8, not very close to 10, you lot should go on.

4. iv

   Set up to find the next digit. Re-create down the next grouping of 3 numbers into the residuum, and draw a pocket-sized vertical line to the left of the resulting number. This will exist the base number for finding the next digit in the solution of your cube root. In this example, this should be the number 2000, which is formed from the remainder 2 of the prior subtraction, with the grouping of three 0s that you pull down.^[iii]

   • To the left of the vertical line, you will be solving the next divisor, as the sum of three split up numbers. Draw the spaces for these numbers by making 3 blank underlines, with plus symbols between them.

5. 5

   Observe the beginning of the next divisor. For the beginning function of the divisor, write downward three hundred times the square of whatsoever is on pinnacle of the radical sign. In this example, the number on top is two, 2^2 is iv, and iv*300=1200. So write 1200 in the outset space. The divisor for this step of the solution will be 1200, plus something that you lot will find next.^[iv]

6. 6

   Find the adjacent number in your cube root solution. Find the next digit of your solution by selecting what you can multiply by the divisor, 1200-something, to then subtract from the rest of 2000. This tin can only exist 1, since 2 times 1200 would exist 2400, which is greater than 2000. Write the number i in the next infinite above the radical sign.^[5]

7. vii

   Determine the rest of the divisor. The divisor for this pace of the solution is made upward of 3 parts. The first function is the 1200 that you already have. You lot need to add two more terms to that to consummate the divisor.^[six]

   • Now calculate 3 times 10 times each of the 2 digits that are in your solution above the radical sign. For this sample problem, that ways 3*ten*2*1, which is threescore. Add together this to the 1200 that you already have to make 1260.
   • Finally, add together the square of the last digit. For this example, that is a i, and 1^2 is still ane. The total divisor is, therefore 1200+60+1, or 1261. Write this to the left of the vertical line.

8. 8

   Multiply and subtract. Complete this round of the solution by multiplying the last digit of your solution - in this case, the number 1 - times the divisor y'all just calculated, 1261. one*1261 =1261. Write this under the 2000, and subtract, to give 739.

9. 9

   Determine whether to keep for more accuracy. After you complete the subtraction portion of each step, you lot need to consider whether your answer is precise enough. For the cube root of ten, after the first subtraction, your cube root was but 2, which is non very precise. Now, subsequently a second round, the solution is 2.1.^[7]

   • You tin check the precision of this consequence past cubing 2.one*2.i*2.ane. The result is 9.261.
   • If y'all believe your result is precise enough, you tin can quit. If yous want a more precise answer, then yous need to proceed with another round.

10. 10

    Discover the divisor for the next round. In this case, for more practice and a more precise answer, echo the steps for another circular, as follows:^[viii]

    • Drop down the next group of iii 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 ${\displaystyle 300*21^{2}}$ , which is 132,300.
    • Select the next digit of your solution so that you can multiply it past 132,300 and take less than the 739,000 of your remainder. A expert option would be 5, since 5*132,300=661,500. Write the digit 5 in the adjacent space above the radical line.
    • Notice three times the prior number to a higher place the radical line, 21, times the last digit y'all merely wrote, 5, times ten. This gives ${\displaystyle 3*21*5*10=3,150}$ .
    • Finally, square the last digit. This is ${\displaystyle five^{2}=25.}$
    • Add the parts of your divisor to get 132,300+3,150+25=135,475.

11. 11

    Multiply the divisor by your solution number. After you have calculated the divisor for this adjacent round and you take expanded your solution past one more digit, go along as follows:

    • Multiply the divisor by the last digit of your solution. 135475*5=677,375.
    • Subtract. 739,000-677,375=61,625.
    • Consider whether the solution of 2.15 is precise enough. Cube it to go ${\displaystyle ii.15*2.15*ii.xv=9.94}$ .

12. 12

    Write down your final answer. The result above the radical is the cube root, authentic at this point to three significant figures. In this example, the cube root of x is ii.xv. Verify that by computing 2.fifteen^3=9.94, which approximates 10. If you need greater accuracy, simply continue the process as long equally you want.

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1. 1

   Apply cube numbers to set upper and lower limits. If you are asked for a cube root of about any number, begin by selecting a perfect cube that is every bit most as possible, without exceeding your target number.

2. ii

   Guess the next digit. The first digit came from your cognition of sure cube numbers. For the next digit, estimate some number betwixt 0 and 9 based on where your target number falls between the two boundary numbers.

   • In the working example, the target of 600 falls about halfway between the boundary numbers of 512 and 729. And so, select 5 for your side by side digit.

3. 3

   Test your estimate past cubing it. Try multiplying out the estimate that you lot are currently working with to run across how shut you get to the target number.

   • In this example, multiply ${\displaystyle 8.5*8.5*8.5=614.i.}$

4. iv

   Adjust your gauge as needed. After cubing your concluding gauge, check where the consequence falls in comparison to your target number. If the result is over the target, you will need to drop your estimate by one or more. If the result is beneath the target, you may need to adjust upwardly until y'all exceed the target.

5. 5

   Estimate the next digit for more precision. Yous volition continue this process of estimating digits from 0 to 9 until your answer is equally precise as you lot want it to be. For each round of estimating, brainstorm by noting how where your latest adding falls between the purlieus numbers.

6. 6

   Proceed to test your gauge and adjust. Equally many times every bit necessary, cube your estimate and see how it compares to your target. Y'all desire to find the numbers that are just below and just above the target number.

7. 7

   Continue as long equally desired for precision. Continue the steps of estimating, comparison and re-estimating equally long as necessary, until your solution is every bit precise as you lot want. Notice that with each decimal identify, your target numbers volition be getting closer and closer to the actual number.

   • For the example of the cube root of 600, when you used two decimal places, 8.43, y'all were abroad from the target by less than ane. If you continue to a third decimal place, you would find that ${\displaystyle 8.434^{3}=599.93}$ , less than 0.1 from the truthful reply.

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1. 1

2. ii

3. iii

   Recognize the pregnant of the long division algorithm. Observe that the method for calculating the cube root works like long division. In long division, y'all find two factors that multiply together to give the product of the number you lot begin with. In the calculation here, the number you are solving for (the number that winds up on elevation of the radical sign) is the cube root. That means that it represents the (10A+B) term. The actual A and B are irrelevant for now, as long as you lot simply recognize the relationship to the respond.^[12]

4. 4

   Review the expanded version. When you look at the expanded polynomial, you lot can run across why the cube root algorithm works. Recognize that the divisor of each step of the algorithm is the sum of four terms that you lot need to calculate and add together. These terms come most as follows:^[13]

   • The first term contains a multiple of 1000. You lot outset a number that could be cubed and stay within the range for the long division for the offset digit. This provides the term 1000A^3 in the binomial expansion.
   • The 2nd term of the binomial expansion has the coefficient of 300. (This actually comes from ${\displaystyle iii*x^{2}}$ .) Recollect that in the cube root calculation, the first digit in each step is multiplied past 300.
   • The 2nd digit in each footstep 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 last digit of each pace is the term B^3.

5. 5

   Run across the precision grow. As you perform the long sectionalization algorithm, each stride that yous complete provides more precision for your answer. For example, the sample problem worked in this commodity is to find the cube root of 10. In the first step, the solution is just two, because ${\displaystyle 2^{three}}$ is close, just less than 10. In fact, ${\displaystyle 2^{iii}=viii}$ . Later on a second round, y'all go the solution of 2.1. When you work this out, ${\displaystyle 2.one^{3}=ix.261}$ , which is much closer to the desired value of x. After a 3rd round, you take two.fifteen, which gives ${\displaystyle 2.xv^{iii}=nine.94}$ . You tin keep working in groups of three digits to become as precise an answer as you need.^[14]

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Add New Question

• Question

  What is the cube root of 27, raised to the 6th power?

  

  The cube root of 27 is 3, and three raised to the sixth power is 729.

• Question

  How exercise I find the fourth root of a number?

  

  The 4th root is the foursquare root of the foursquare root.

• Question

  What is the cube root of 24?

  

  This question might be simplest to solve using the repeated estimation method described in the article. You should know that the cube root must be nearly 3, since three cubed is 27, and 24 is very close to 27. So yous could first approximate that the cube root is 2.eight. You'll notice that 2.8 cubed is 21.952, and 2.9 cubed is 24.389. Thus, the cube root of 24 is a scrap more than than 2.8 but less than 2.9. The target of 24 is very nearly to 24.389, so choose a number closer to 2.ix than 2.eight. Estimate two.88. You'll notice that 2.88 cubed is 23.888, but ii.89 cubed is 24.138. And then the solution is between ii.88 and 2.89, nigh exactly in the middle. Then the adjacent estimate would be 2.885. Then continue as long as y'all like for precision.

• Question

  The cube root of 9 is 2.080, while our answer using that method is 2.082 (iii decimal places). We checked many times and it still does not give the right respond. Does it mean in that location are limits in using this?

  

  No. The algorithm works merely fine. Yous should check your sources and recheck your work. This could be a question of precision. When you begin by saying "the cube root of ix is 2.080," what is your source for that? 2.080 cubed = eight.9989. When written without decimals, this means ii.080 cubed is 9. ii.081 cubed = ix.0119. When written without decimals, this means two.081 cubed is 9. 2.082 cubed = 9.0249. When written without decimals, this means 2.082 cubed is nine. In fact, yous can go as loftier as ii.xi, cubed, and still say that the result is 9.

• Question

  What is the cube root of 0.04?

  

  The process described in this article works for solving the cube roots of numbers less than one also as numbers greater than one. You will gear up the problem in just the aforementioned way. Remember to write the numbers in groups of 3 digits, starting time at the decimal point. In this case, you will write 0.04 equally 0.040 000 000. Each group of three digits will consequence in i digit of your solution. The work is likewise lengthy to write out here. The solution, to three decimal places, is 0.341.

• Question

  How is determining the foursquare root of a number like to determining its cube root?

  

  Squaring is multiplying a number by itself. Cubing is multiplying the square by the base number again.

• Question

  What is the cube root of 1331?

  

  Co-ordinate to Pascal'south triangle coefficients, the starting time line is 1 for 0th power,2d line is11,3rd line is 121 for squared and the fourth line is 1331 for the cube of eleven.In reverse,the cube root of 1331 is 11.

• Question

  How do I calculate the cube root of 0.001 without using a calculator?

  

  This answer simply works simply and cleanly because you have selected .001. This is non so difficult, but again merely because it'south this number. three√.001=x is equal to 10^3 = .001, and since we are merely talking virtually .001, consider that .001 is actually one thousandth then consider what whole x^iii would give you lot 1,000. This means your answer would exist yard's cubed root. 10 x x x ten = k, but nosotros accept the decimal, pregnant 10 but 1 tenth would be the cubed root of 1 thousandth, or .one x .1 ten .1= .001 and the inverse 3√.001=.one.

• Question

  What's the method of finding cube root of 7.97 using approximation?

  

  Starting time with knowing that two cubed is 8. From at that place, you lot already know that the cube root of 7.97 will be just under ii, since seven.97 is just under 2 cubed. With that knowledge, you can guess ane.99. This gives 7.88, which isn't shut enough. Attempt one.999. Yous get vii.988, which is a little likewise loftier. Endeavour i.995. Yous get 7.94, which is too low. Try one.998. Yous go 7.976, which is very close. If you want to be more specific, you tin endeavour numbers betwixt 1.997 and 1.998, similar 1.9976, and y'all tin can go along getting closer and closer. The bones idea is to try a number, then encounter if that number is also high or likewise depression. If information technology's too low, increase it. If it's too high, lower information technology.

• Question

  How do I notice foursquare roots?

  

  There are several guides on wikiHow that can assistance you lot find square roots, like How to Calculate a Square Root by Hand and How to Find a Square Root Without a Figurer.

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Article Summary X

To calculate cube root past paw, choose a perfect cube that is as close to the answer as possible, write it downwardly, and subtract your approximate from the original number. For example, you could estimate that the square root of 30 was 3. Nonetheless, 3 cubed is 27, so you would write down 3 as the kickoff part of your respond with a remainder of 3. Then, judge what cubed would fit into the remainder and subtract it, likewise. Repeat that process until you've reached your desired accurateness. Keep reading to learn how to find cube roots through long division.

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