Sum of Indices of Real Number
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Theorem
Let $r \in \R_{> 0}$ be a (strictly) positive real number.
Positive Integers
Let $n, m \in \Z_{\ge 0}$ be positive integers.
Let $r^n$ be defined as $r$ to the power of $n$.
Then:
- $r^{n + m} = r^n \times r^m$
Integers
Let $n, m \in \Z$ be integers.
Let $r^n$ be defined as $r$ to the power of $n$.
Then:
- $r^{n + m} = r^n \times r^m$
Rational Numbers
Let $x, y \in \Q$ be rational numbers.
Let $r^x$ be defined as $r$ to the power of $n$.
Then:
- $r^{x + y} = r^x \times r^y$
Sources
- 1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.2$ Operations with Real Numbers: $(6)$