Exponent Combination Laws/Product of Powers/Proof 2
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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:
- $a^x a^y = a^{x + y}$
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Proof
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:
\(\ds a^{x + y}\) | \(=\) | \(\ds a^{\ds \paren {\lim_{n \mathop \to \infty} x_n + \lim_{n \mathop \to \infty} y_n} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a^{\ds \paren {\lim_{n \mathop \to \infty} \paren {x_n + y_n} } }\) | Sum Rule for Real Sequences | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} a^{x_n + y_n}\) | Sequential Continuity is Equivalent to Continuity in the Reals | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \paren {a^{x_n} a^{y_n} }\) | Sum of Indices of Real Number: Rational Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\lim_{n \mathop \to \infty} a^{x_n} } \paren {\lim_{n \mathop \to \infty} a^{y_n} }\) | Product Rule for Real Sequences | |||||||||||
\(\ds \) | \(=\) | \(\ds a^x a^y\) | Sequential Continuity is Equivalent to Continuity in the Reals |
$\blacksquare$