Binomial theorem
The binomium with a rational exponent can be written as
in which m and n are natural numbers.
Explanation
The binomial coefficients for the terms in the development of the binomial theorem, with a rational exponent, can be written as
The upper fraction m/n is the exponent of the binomial theorem, the lower number k is the running number of the term in the outcome. A number arises from the sum of the number just below it and to the left of it. You can see it in the penultimate row at
All rows have an infinite number of terms.
m/n 0 1 2 3 4 5 k → ∞ ··· ··· ··· ··· ··· ··· ··· ··· 5/2 ··· 3/2 ··· 1/2 ··· 0/2 ◦ ◦ ◦ ◦ ◦ ··· −1/2 ··· −3/2 ··· ··· ··· ··· ··· ··· ··· ··· ···
For rational exponents we have the formula
Example 1
The mass-energy relation is determined by the expression