Exponent Laws: Product, Quotient, and Power Rules

Three shortcuts for combining exponents without expanding everything out.

🧱 See It

Write out 2³ × 2² fully: (2×2×2) × (2×2) is five 2's multiplied together — 2⁵. Notice the exponents 3 and 2 just added to make 5.

✏️ Draw It

Written outShortcut
(2×2×2) × (2×2)2³ × 2² = 2⁵

Same base, multiplying → add the exponents.

🔢 Write It

RuleFormula
Productxm × xn = xm+n
Quotientxm ÷ xn = xm−n
Power(xm)n = xm×n
All three rules only work when the base is the SAME on both sides. x³ × y² cannot be combined — different bases, no shortcut.

💡 Worked Examples

Example 1 — Product rule: x⁴ × x³

  1. Same base, multiplying → add exponents: 4 + 3 = 7
  2. Answer: x⁵
  3. Check with x=2: 2⁴ × 2³ = 16 × 8 = 128, and 2⁵ = 128 — matches.

Example 2 — Quotient rule: x⁵ ÷ x²

  1. Same base, dividing → subtract exponents: 5 − 2 = 3
  2. Answer: x³
  3. Check with x=2: 2⁵ ÷ 2² = 32 ÷ 4 = 8, and 2³ = 8 — matches.

🧠 Quick Reference

Multiplying same base → ADD exponents. Dividing same base → SUBTRACT exponents. A power raised to another power → MULTIPLY exponents. Always check the bases match before using any of these.
👪 For Parents & Tutors

BC Curriculum (Level 4 / Grade 9, Exponents): "Laws including zero exponents, product rule, quotient rule, power rules."

Watch for: applying a rule to two DIFFERENT bases (e.g. trying to combine x³ × y²) — all three rules require the same base on both sides. Have him check "is the base the same?" as an explicit first question before reaching for any rule. A second trap: mixing up which operation (add, subtract, or multiply the exponents) belongs to which rule — the numeric verification shown in each worked example is a good habit to keep, since it catches a wrong rule immediately.

Comes up again: constantly throughout this level's Polynomials pages, where multiplying terms uses the product rule on nearly every line.