Repeating Multiplication
An exponent tells you how many times to multiply a base by itself. The Product Rule explains what happens when you multiply two powers with the same base.
\[x^a \cdot x^b = x^{a+b}\]
Worked Examples
Example 1: Basic Variables
Simplify: \(x^3 \cdot x^4\)
- Think: \((x \cdot x \cdot x) \cdot (x \cdot x \cdot x \cdot x)\). That's 7 \(x\)'s.
- Rule: \(x^{3+4} = x^7\).
- Result: \(x^7\)
Example 2: Multiple Bases
Simplify: \((2x^2y)(5x^3y^4)\)
- Multiply coefficients: \(2 \times 5 = 10\).
- Add \(x\) exponents: \(x^{2+3} = x^5\).
- Add \(y\) exponents: \(y^{1+4} = y^5\). (Note: \(y\) is secretly \(y^1\)).
- Result: \(10x^5y^5\)
Example 3: Negative Exponents
Simplify: \(x^5 \cdot x^{-2}\)
- Add: \(5 + (-2) = 3\).
- Result: \(x^3\)
The Bridge to Quantum Mechanics
In the "Many-Body Problem," where we have multiple electrons, the total wavefunction is often a product of individual wavefunctions: \(\Psi = \psi_1 \psi_2 \dots\). If these wavefunctions involve exponential factors (which they almost always do, like \(e^{ikx}\)), the Product Rule allows us to combine them into a single exponential. This is the only way physicists can handle the math of complex atoms with many electrons.