The extraction of roots of numbers higher than cube roots, was, according to the writings of Omar Khayyam, an achievement of Muslim scholars. Omar Khayyam wrote in his Treatise on Demonstration of Problems of Algebra:-
From the Indians one has methods for obtaining square and cube roots, methods which are based on knowledge of individual cases, namely the knowledge of the squares of the nine digits 1^{2}, 2^{2} , 3^{2} (etc.) and their respective products, i.e. 2 3 etc. We have written a treatise on the proof of the validity of those methods and that they satisfy the conditions. In addition we have increased their types, namely in the form of the determination of the fourth, fifth, sixth roots up to any desired degree. No one preceded us in this and those proofs are purely arithmetic, founded on the arithmetic of The Elements.
In fact al-Kashi had extracted the fifth root of 44 240 899 506 176. For those who are not able to do this today, we inform the reader that the fifth root is 536 (which we admit we computed using Maple!).
JOC/EFR March 2006
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