section_path: Computing principal roots/Digit-by-digit calculation of principal roots of decimal (base 10) numbers
section_name: Digit-by-digit calculation of principal roots of decimal (base 10) numbers
paragraph_text: Digit-by-digit calculation of principal roots of decimal (base 10) numbers
  
  Building on the digit-by-digit calculation of a square root, it can be seen that the formula used there, x (20 p + x) ≤ c {\displaystyle x(20p+x)\leq c}, or x 2 + 20 x p ≤ c {\displaystyle x^{2}+20xp\leq c}, follows a pattern involving Pascal's triangle. For the n th root of a number P (n, i) {\displaystyle P(n,i)} is defined as the value of element i {\displaystyle i} in row n {\displaystyle n} of Pascal's Triangle such that P (4, 1) = 4 {\displaystyle P(4,1)=4}, we can rewrite the expression as ∑ i = 0 n − 1 10 i P (n, i) p i x n − i {\displaystyle \sum _{i=0}^{n-1}10^{i}P(n,i)p^{i}x^{n-i}}. For convenience, call the result of this expression y {\displaystyle y}. Using this more general expression, any positive principal root can be computed, digit-by-digit, as follows.
  
  Write the original number in decimal form. The numbers are written similar to the long division algorithm, and, as in long division, the root will be written on the line above. Now separate the digits into groups of digits equating to the root being taken, starting from the decimal point and going both left and right. The decimal point of the root will be above the decimal point of the radicand. One digit of the root will appear above each group of digits of the original number.
  
  Beginning with the left-most group of digits, do the following procedure for each group: