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The cube root of 456533 is 77. This process can be extended to find cube roots that are 3 digits long, by using arithmetic modulo 11. [3] These types of tricks can be used in any root where the order of the root is coprime with 10; thus it fails to work in square root, since the power, 2, divides into 10. 3 does not divide 10, thus cube roots work.
A natural number is a sociable Dudeney root if it is a periodic point for ,, where , = for a positive integer , and forms a cycle of period . A Dudeney root is a sociable Dudeney root with k = 1 {\displaystyle k=1} , and a amicable Dudeney root is a sociable Dudeney root with k = 2 {\displaystyle k=2} .
Examples and several cube root digit schedules are given. 5th root, 7th root, and nth root general formula only hinted at by the binomial expansion of that power. Page 340, Vedic Mathematics, 1965, 1978. Larry R. Holmgren 20:03, 3 March 2007 (UTC) How much space is appropriate for examples? Larry R. Holmgren 19:02, 4 March 2007 (UTC)
A square root of a number x is a number r which, when squared, becomes x: =. Every positive real number has two square roots, one positive and one negative. For example, the two square roots of 25 are 5 and −5. The positive square root is also known as the principal square root, and is denoted with a radical sign:
For example, the square root of a number is the same as raising the number to the power of and the cube root of a number is the same as raising the number to the power of . Examples are 4 = 4 1 2 = 2 {\displaystyle {\sqrt {4}}=4^{\frac {1}{2}}=2} and 27 3 = 27 1 3 = 3 {\displaystyle {\sqrt[{3}]{27}}=27^{\frac {1}{3}}=3} .
The principal cube root is the cube root with the largest real part. In the case of negative real numbers, the largest real part is shared by the two nonreal cube roots, and the principal cube root is the one with positive imaginary part. So, for negative real numbers, the real cube root is not the principal cube root. For positive real numbers ...
For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128. The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7.
The extraction of decimal-fraction approximations to square roots by various methods has used the square root of 7 as an example or exercise in textbooks, for hundreds of years. Different numbers of digits after the decimal point are shown: 5 in 1773 [ 4 ] and 1852, [ 5 ] 3 in 1835, [ 6 ] 6 in 1808, [ 7 ] and 7 in 1797. [ 8 ]