Exponent Combination Laws/Product of Powers
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
| \(\ds a^{x + y}\) | \(=\) | \(\ds \map \exp {\paren {x + y} \ln a}\) | Definition of Power to Real Number | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \exp {x \ln a + y \ln a}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map \exp {x \ln a} \, \map \exp {y \ln a}\) | Exponential of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds a^x a^y\) | 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:
| \(\ds a^{x + y}\) | \(=\) | \(\ds a^{\ds \paren {\lim_{n \mathop \to \infty} x_n + \lim_{n \mathop \to \infty} y_n} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds a^{\ds \paren {\lim_{n \mathop \to \infty} \paren {x_n + y_n} } }\) | 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 |
$\blacksquare$
Also known as
The Exponent Combination Laws is also known as:
- the Laws of Exponents
- the Laws of Indices.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Laws of Exponents: $7.1$
- 1972: Murray R. Spiegel and R.W. Boxer: Theory and Problems of Statistics (SI ed.) ... (previous) ... (next): Chapter $1$: Scientific Notation
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): exponent (index)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): exponent (index)
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 13$: Laws of Exponents: $13.1.$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): index (indices) (i)