 Al-Khashis, the calculator's key, included an algorithm for extracting fifth roots. We know Awawafa and Al-Khyami wrote treatises on extracting higher-order roots, but these have not survived, so Al-Khashis is the oldest work still in existence. Al-Khashis' algorithm is meant to be implemented on paper where a permanent record of the computations could be kept. Unfortunately, trying to do so on a small screen would render the numbers too small to see. So we'll perform the computations and replace the numbers as we find them. To illustrate the process, let's find the fifth root of this number. So first, because it's the fifth root, Al-Khashis breaks the number into five-digit sets, which he calls cycles. Now, there's one full cycle of five and one partial cycle, and this tells us that the root has two whole-numbered digits. Next, the page is going to be divided into sections. Since we're finding a fifth root, it's convenient to form five sections. The first section is the one that contains the number. Al-Khashis calls this the row of the number. And since we're trying to extract the fifth root, we can think of our number as the fifth power of something, and that means the row below it is the fourth power, which Al-Khashis called the row of the square-square. So that we have the row of the cube, the row of the square, the row of the root, and then finally the row of the result. So let's find the fifth root, and so we'll begin by guessing the first digit of the root. And in many ways, this is the most complicated part. The key requirement is that we must be able to perform a sequence of operations and subtract the final result from the number. And the best way to explain that statement is to try and work the process. So let's guess one as our first digit. Now we'll add our guess to the row of the root, and since there's nothing there, that just becomes one. And now we're going to perform a sequence of multiplications and additions, which will always work as follows. The guess is going to be multiplied by a number and added to the next row up. So our guess is multiplied by the row of the root and added to the number in the row of the square. Again, since there's nothing there, then it's the only number in the row of the square. Then guess times number in row of the square and add to the row of the cube. Guess times row of the cube and add to the row of the square-square. Now notice we're at the top of the table, and so now something is going to happen that's a little bit different. We're going to multiply our guess by the number in the row of the square-square and set down in the row of the number, and this time we're going to subtract. Now it's worth noting that here we progressed from the bottom all the way to the top. What we're going to do now is we're going to repeat the process, but we're going to stop one row short. A process that Al-Qashir refers to as once up to the row of the square-square. So again, we'll add our guess to the number in the row of the root. And now our guess is multiplied by the number in the row of the root and added to the number in the row of the square. Then our guess is multiplied by the number in the row of the square and added to the number in the row of the cube. And our guess is multiplied by the number in the row of the cube and added to the number in the row of the square-square. But wait, we're not done. We're going to go once up to the row of the cube. And so again, we add our guess to the number in the row of the root. Then guess times number of the row of the root and add to the number in the row of the square. Then guess times number in the row of the square and add to the number in the row of the cube. And now we go once up to the row of the square. So again, guess added to the number in the row of the root. Then guess times number in the row of the root and add to the number in the row of the square. One last time, once up to the row of the root, where we don't actually go very far, we add our guess to the number in the row of the root. And now we're going to shift our digits. To keep everything straight, we'll put down some placeholder lines. The digits in the row of the square square are shifted one place. The digits in the row of the cube are shifted two places. The digits in the row of the square are shifted three places. The digits in the row of the root are shifted four places. And we can get rid of our placeholding lines as long as we keep in mind that there are placeholding zeros. And now we lather, rinse, repeat. The important thing is that when we repeat that sequence of add, multiply, and add to the next row, when we get to the top, we have to be able to perform that last subtraction. And so we guess the next digit of the root, and it turns out that eight will work. So we add our guess to the row of the root. Our guess is multiplied by the row of the root and added to the number in the row of the square. Our guess is multiplied by the number of the row of the root and added to the number in the row of the cube. Our guess is multiplied by the number of the row of the cube and added to the number in the row of the square. Our guess is multiplied by the number of the row of the square, and now we set it down at the row of the number, and then we subtract. And we go through our procedure again once up to the row of the square. So we add our guess to the number in the row of the root. Guess times number and add. Guess times number and add. Guess times number and add and stop. Now once up to the row of the cube. Add guess to number. Then guess times number and add. Then guess times number and add. And we happy few who get this far will go once up to the row of the square. And once up to the row of the root. And our fifth root so far is eighteen. Now Akashi probably knew that he could continue the procedure and obtain additional decimal digits of the root. Instead he approximated the fractional part of the root as follows. The numerator of the fraction is the remaining part of the number. For the denominator, Akashi took the sum of the numbers in the rows of the square square through the root plus one. And so our fifth root is approximately