NapkinCalc

Algebra 1B — Systems, Exponents & Factoring

Exponent rules

continues from lesson 1 — values defined earlier in the course stay live here

ELI5: exponents are repeated multiplication, and three rules cover almost everything:

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

Don't memorize blindly — verify with a real base:

base=2base = 2 try 7 or 0.5 too
✓ pass abs(base3⋅base4−base7)<10−9\mathrm{abs}\left(base^{3} \cdot base^{4} - base^{7}\right) < 10^{-9} product rule: 2³·2⁴ = 2⁷
✓ pass abs((base3)2−base6)<10−9\mathrm{abs}\left(\left(base^{3}\right)^{2} - base^{6}\right) < 10^{-9} power rule: (2³)² = 2⁶
✓ pass abs(base−2−1base2)<10−9\mathrm{abs}\left(base^{-2} - \frac{1}{base^{2}}\right) < 10^{-9} negative exponent is a reciprocal

Real-world hook: exponents run scientific notation (6.02×10²³), compound interest, and every "doubles every N days" growth story.

Try it yourself: 2⁵ · 2³ = 2ⁿ — what is the exponent n? (Product rule.)

nyou=8n_{you} = 8 ✏️ Your turn: replace 0 with the exponent n so that 2ⁿ = 2⁵·2³.
✓ pass abs(2nyou−25⋅23)<0.000001\mathrm{abs}\left(2^{n_{you}} - 2^{5} \cdot 2^{3}\right) < 0.000001 green when 2ⁿ equals 2⁵·2³