Exponent Combination Laws/Product of Powers

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Theorem

Let $a \in \R_{> 0}$ be a positive real number.

Let $x, y \in \R$ be real numbers.

Let $a^x$ be defined as $a$ to the power of $x$.


Then:

$a^x a^y = a^{x + y}$


Proof 1

\(\displaystyle a^{x + y}\) \(=\) \(\displaystyle \map \exp {\paren {x + y} \ln a}\) Definition of Power to Real Number
\(\displaystyle \) \(=\) \(\displaystyle \map \exp {x \ln a + y \ln a}\)
\(\displaystyle \) \(=\) \(\displaystyle \map \exp {x \ln a} \, \map \exp {y \ln a}\) Exponential of Sum
\(\displaystyle \) \(=\) \(\displaystyle a^x a^y\) Definition of Power to Real Number

$\blacksquare$


Proof 2

Let $x, y \in \R$.


From Rational Sequence Decreasing to Real Number, there exist rational sequences $\sequence {x_n}$ and $\sequence {y_n}$ converging to $x$ and $y$, respectively.


Then, since Power Function on Strictly Positive Base is Continuous: Real Power:

\(\displaystyle a^{x + y}\) \(=\) \(\displaystyle a^{\displaystyle \paren {\lim_{n \mathop \to \infty} x_n + \lim_{n \mathop \to \infty} y_n} }\)
\(\displaystyle \) \(=\) \(\displaystyle a^{\displaystyle \paren {\lim_{n \mathop \to \infty} \paren {x_n + y_n} } }\) Sum Rule for Real Sequences
\(\displaystyle \) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} a^{x_n + y_n}\) Sequential Continuity is Equivalent to Continuity in the Reals
\(\displaystyle \) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} \paren {a^{x_n} a^{y_n} }\) Sum of Indices of Real Number: Rational Numbers
\(\displaystyle \) \(=\) \(\displaystyle \paren {\lim_{n \mathop \to \infty} a^{x_n} } \paren {\lim_{n \mathop \to \infty} a^{y_n} }\) Product Rule for Real Sequences
\(\displaystyle \) \(=\) \(\displaystyle a^x a^y\) Sequential Continuity is Equivalent to Continuity in the Reals

$\blacksquare$


Sources