Properties of powers with a natural exponent

Powers with a natural exponent have five properties. 1) aᵐ · aⁿ = aᵐ⁺ⁿ: when multiplying powers with the same base, the exponents are added. 2) aᵐ : aⁿ = aᵐ⁻ⁿ (m > n, a ≠ 0): when dividing, the exponents are subtracted. 3) (aᵐ)ⁿ = aᵐⁿ: raising a power to a power multiplies the exponents. 4) (ab)ⁿ = aⁿbⁿ: raising a product to a power raises each factor to that power. 5) (a/b)ⁿ = aⁿ/bⁿ (b ≠ 0): the numerator and denominator are raised separately. These properties follow from the definition of a power: for instance, a² · a³ = (a · a) · (a · a · a) = a⁵. We also use them in the reverse direction: 2³ · 5³ = (2 · 5)³ = 1000.