13th root 13th root Extracting the 13th root of a number is a famous category for the mental calculation world records. The challenge consists of being given a large number (possibly over 100 digits) and asked to return the number that, when taken to the 13th power, equals the given number. For example, the 13th root of 8,192 is 2 and the 13th root of 96,889,010,407 is 7. Extracting the 13th root has certain properties. One is that the 13th root of a number is much smaller: a 13th root will have approximately 1/13th the number of digits. Thus, the 13th root
Square root of a 2 by 2 matrix use the simplified formula where "a" = ±√"A" and "d" = ±√"D"; which, for the various sign choices, gives four, two, or one distinct matrices, if none of, only one of, or both "A" and "D" are zero, respectively. Because it has duplicate eigenvalues, the 2×2 identity matrix formula_10 has infinitely many symmetric rational square roots given by where is any Pythagorean triple—that is, any set of positive integers such that formula_12 In addition, any non-integer, irrational, or complex values of "r, s, t" satisfying formula_13 give square root matrices. The identity matrix also has infinitely many non-symmetric square roots.