{
 "source": "https://peabody.kids/flashcards/math-exponent-rules",
 "id": "math-exponent-rules",
 "title": "Exponent Rules",
 "category": "math",
 "subject": "Math",
 "levels": [
  "6-8",
  "9-12"
 ],
 "description": "The laws of exponents, with worked examples including zero, negative and fractional exponents.",
 "tags": [
  "exponents",
  "powers",
  "algebra"
 ],
 "frontLabel": "Rule",
 "backLabel": "Result",
 "cards": [
  [
   "Product rule",
   "aᵐ · aⁿ = aᵐ⁺ⁿ",
   "Same base: add exponents"
  ],
  [
   "Quotient rule",
   "aᵐ / aⁿ = aᵐ⁻ⁿ",
   "Same base: subtract exponents"
  ],
  [
   "Power of a power",
   "(aᵐ)ⁿ = aᵐⁿ",
   "Multiply exponents"
  ],
  [
   "Power of a product",
   "(ab)ⁿ = aⁿbⁿ"
  ],
  [
   "Power of a quotient",
   "(a/b)ⁿ = aⁿ/bⁿ"
  ],
  [
   "Zero exponent",
   "a⁰ = 1, for a not zero"
  ],
  [
   "Negative exponent",
   "a⁻ⁿ = 1/aⁿ"
  ],
  [
   "Fractional exponent",
   "a^(1/n) = the nth root of a"
  ],
  [
   "Rational exponent",
   "a^(m/n) = (nth root of a)^m"
  ],
  [
   "Exponent of 1",
   "a¹ = a"
  ],
  [
   "2³ · 2⁴",
   "2⁷ = 128"
  ],
  [
   "5⁶ / 5²",
   "5⁴ = 625"
  ],
  [
   "(3²)³",
   "3⁶ = 729"
  ],
  [
   "(2x)³",
   "8x³"
  ],
  [
   "(x²)⁴",
   "x⁸"
  ],
  [
   "x³ · x⁵",
   "x⁸"
  ],
  [
   "x⁹ / x³",
   "x⁶"
  ],
  [
   "(−2)²",
   "4"
  ],
  [
   "−2²",
   "−4",
   "The exponent applies only to the 2"
  ],
  [
   "(−2)³",
   "−8"
  ],
  [
   "10⁰",
   "1"
  ],
  [
   "0⁰",
   "Usually left undefined (or treated as 1 in some settings)"
  ],
  [
   "2⁻³",
   "1/8"
  ],
  [
   "10⁻²",
   "0.01"
  ],
  [
   "4^(1/2)",
   "2"
  ],
  [
   "8^(1/3)",
   "2"
  ],
  [
   "27^(2/3)",
   "9"
  ],
  [
   "16^(3/4)",
   "8"
  ],
  [
   "9^(−1/2)",
   "1/3"
  ],
  [
   "Is (a + b)² equal to a² + b²?",
   "No. (a + b)² = a² + 2ab + b²"
  ],
  [
   "Is aᵐ + aⁿ = aᵐ⁺ⁿ?",
   "No. The product rule works for multiplying, not adding"
  ],
  [
   "Negative base with even power",
   "The result is positive"
  ],
  [
   "Negative base with odd power",
   "The result is negative"
  ],
  [
   "Scientific notation of 45,000",
   "4.5 × 10⁴"
  ],
  [
   "Scientific notation of 0.0007",
   "7 × 10⁻⁴"
  ],
  [
   "(3 × 10²)(2 × 10³)",
   "6 × 10⁵"
  ],
  [
   "Square root as an exponent",
   "√a = a^(1/2)"
  ],
  [
   "Cube root as an exponent",
   "∛a = a^(1/3)"
  ],
  [
   "Reciprocal of x⁻²",
   "x²"
  ],
  [
   "Simplify (x³y²)²",
   "x⁶y⁴"
  ],
  [
   "Simplify x⁻²y³ with positive exponents",
   "y³/x²"
  ]
 ]
}