To extend the application of exponentiation to the use of irrational numbers as exponents, we must define exponentiation by irrational numbers. The basic idea is to obtain a sequence of rational numbers that approaches the irrational exponent, to compute the corresponding powers, and to define the power to be the limit that is approached by the powers with rational exponents.
For example, to raise 2 to the power √2, we approximate the irrational exponent by rational numbers that approach the value of the irrational exponent.
This sequence eventually converges to:√2 2x 2√2 ------------------------------------------- 1 21 2 1.4 21.4 2.639016... 1.41 21.41 2.657372... 1.414 21.414 2.664750... 1.4142 21.4142 2.665119...
The laws of exponents for irrational exponents are identical as those that apply to rational exponents.2√2 = 2.665144...