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How did Islamic mathematicians extend root extraction beyond square and cube roots?

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arithmetic A page from a first-grade workbook typical of “new math” might state: “Draw connecting lines from triangles in the first set to triangles in the second set. Are the two sets equivalent in number?”

Islamic mathematicians significantly advanced arithmetic by extending root-extraction procedures beyond the square and cube roots known to the Greeks and Indians, eventually developing methods to extract roots of any desired degree.12

A foundational step in this development was the work of the 10th-century mathematician al-Karajī, who derived the binomial theorem for whole-number exponents using mathematical induction. This insight into the expansion of powers provided the theoretical basis for higher-degree root extraction.1

While specific treatises on the topic by scholars such as Abū al-Wafā’ and Omar Khayyam have been lost, historical evidence confirms that these mathematicians successfully generalized the process. Omar Khayyam, in particular, is credited with solving the general problem of extracting roots of arbitrary high degree, further cementing the Islamic contribution to the evolution of algebraic methods.12