Cube Roots by Hand


Calculator Appreciation Day

Few by-hand calculations will make you appreciate your hand-held magic number crunching machine more than working out square roots or cube roots by hand. We recently looked at square roots by hand » and as you might expect, cube roots by hand require a very similar approach, with a slight increase in mechanics and difficulty.We'll walk through the process with an example. Let's take the cube root of $500$ to two decimal places. Since $7^3 = 343$ and $8^3 = 512$, we should expect our answer to be $7.[\mathrm{something}]$. Once again, we will set up something that looks a lot like long division, including extra $0$ placeholders for decimal places.Step 1 - SetupMark the decimal place and "chunk off" digits in groups of three in each direction. It's ok if the last group has one, two, or three digits in it. In our walkthrough example our first group ($500$) happens to have three digits.Note that we did something similar for the square roots by hand » process, except that we broke up our digits into chunks of two.Step 2 - First DigitThe first digit is obtained by "brute force" - that is, determine the biggest integer whose cube is smaller than the first group.Our first group is $500$, and the largest cube less than $500$ is $7^3 = 343$.Step 3 - Subtract and Bring DownWrite the cubed result underneath the $500$ and subtract - nearly identical to the mechanics you remember from old school long division.Then bring down the next three digits - once again reminding us of how long division works.Step 4 - Seek Next DigitThis is the awkward and challenging step.Let's call your answer so far (everything on top of the radicand) $q$, ignoring decimals. So far, our $q$ is $7$. At the same time, call your current remainder $r$, which is the number you're left with after you did the prior "bring down" step. In our example, our current $r$ is $157,000$.The next digit in the answer will be the digit $d$ such that$$(300q^2 + 30qd + d^2) \times d \leq r$$Ok, so it's not the simplest expression to work with, but fortunately we don't have to algebraically solve for $d$ - we can guess and check. This will also ring familiar to division. When you think about it, we also had to do some guess and check in the old-school long division process in order to find the next digit.In our example, we need to guess and check to find $d$ such that$$(300(7)^2 + 30(7)(d) + d^2) \times d \leq 157,000$$$$(14,700 + 210d + d^2) \times d \leq 157,000$$After a few guesses we confirm that $9$ is that number, since the expression above is equal to $150,039$ when $d$ is $9$. Place $9$ on top above the next chunk of three numbers.Step 5 - Rinse and RepeatIt is now time to repeat steps 3 and 4 until you have achieved your desired level of accuracy. So far we have an answer accurate to one decimal place. Let's get one more.To repeat Step 3 (subtract / bring down), use the last result you obtained from that awful $d$ expression. Our last $d$ was $9$, and we got $150,039$. Jot it down, subtract, and bring down.To repeat Step 4 (find next digit), seek out the next $d$ using your updated values of $q$ and $r$ ($79$ and $6,961,000$, respectively).This is entirely disgusting and arguably painstaking without a calculator. Cube roots by hand isn't an easy task!$$(300(79)^2 + 30(79)d + d^2) \times d \leq 6,961,000$$For $q=79$, $(300q^2 +30qd + d^2)\times d$ is equal to $5,638,257$ when $d$ is $3$, and we cannot make it higher without exceeding $6,961,000$. Therefore the next digit in our desired solution is $3$.That's as far as we'll go for this example - and grabbing a calculator to check confirms that we are correct:$$\sqrt[3]{500} \approx 7.937005$$Try one yourself! Can you reproduce the following example for $\sqrt[3]{2700}$?