NapkinCalc

Algebra 1

Exponent rules

continues from lesson 3 — values defined earlier in the course stay live here

Three rules cover almost everything:

  • Product: aᵐ · aⁿ = aᵐ⁺ⁿ — multiplying stacks the exponents
  • Power: (aᵐ)ⁿ = aᵐⁿ
  • Negative: a⁻ⁿ = 1/aⁿ — a negative exponent is a flip, not a negative number

Don't memorize — verify. The checks below test the rules with a real number:

a=2a = 2 any base you like — try 7 or 0.5
✓ pass abs(a3⋅a4−a7)<10−9\mathrm{abs}\left(a^{3} \cdot a^{4} - a^{7}\right) < 10^{-9} product rule: a³·a⁴ = a⁷
✓ pass abs((a3)2−a6)<10−9\mathrm{abs}\left(\left(a^{3}\right)^{2} - a^{6}\right) < 10^{-9} power rule: (a³)² = a⁶
✓ pass abs(a−2−1a2)<10−9\mathrm{abs}\left(a^{-2} - \frac{1}{a^{2}}\right) < 10^{-9} negative exponent is a reciprocal

Where next: Algebra 2 picks up with quadratics — equations where x appears squared and lines bend into parabolas.