Laws of Exponents

The exponent laws all come from counting factors.

x³ · x⁴ means (xxx)(xxxx) = x⁷. So multiplying adds exponents.
x⁵/x² cancels two x's and leaves x³. So dividing subtracts exponents.
(x³)² means x³ · x³ = x⁶. So a power of a power multiplies exponents.
(xy)³ = x³y³. The exponent reaches every factor inside.

The critical condition: the bases must match. x³ · y⁴ cannot be combined at all.

And the biggest trap: x³ + x⁴ does not simplify. These laws are for multiplying and dividing, never for adding.

For a ≠ 0 and integers m, n:

aᵐ·aⁿ = aᵐ⁺ⁿ aᵐ/aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ (ab)ⁿ = aⁿbⁿ (a/b)ⁿ = aⁿ/bⁿ

Each is proved for positive integers by counting factors, then *extended* to zero and negative exponents by requiring the rules to keep holding. Applying the quotient rule to aⁿ/aⁿ forces a⁰ = 1; applying it to a⁰/aⁿ forces a⁻ⁿ = 1/aⁿ. Negative exponents are therefore not a new idea but the unique consistent extension.

No rule exists for aᵐ + aⁿ, because addition of powers does not factor into a single power. The most that can be done is factoring out the smaller: x³ + x⁴ = x³(1 + x).

Note also that (a + b)ⁿ ≠ aⁿ + bⁿ. Exponentiation distributes over multiplication, never over addition — the correct expansion requires the binomial theorem.