text: The properties of fractional exponents also follow from the same rule. For example, suppose we consider  and ask if there is some suitable exponent, which we may call , such that . From the definition of the square root, we have that . Therefore, the exponent  must be such that . Using the fact that multiplying makes exponents add gives . The  on the right-hand side can also be written as , giving . Equating the exponents on both sides, we have . Therefore, , so .
  The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices.
  Exponentiation is used extensively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography.
  Etymology
  The term exponent originates from the Latin exponentem, the present participle of exponere, meaning "to put forth". The term power () is a mistranslation of the ancient Greek δύναμις (dúnamis, here: "amplification") used by the Greek mathematician Euclid for the square of a line, following Hippocrates of Chios.
  History
  Antiquity