{
  "title": "Negative and Fractional Indices - Grade 9 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "What is 5⁻¹? 😯", "options": {"A": "-5", "B": "0.5", "C": "0.2", "D": "1"}, "correct_answer": "C"},
    {"id": 2, "difficulty": "easy", "question": "Simplify: 2⁻² = ? 🔽", "options": {"A": "-4", "B": "0.25", "C": "4", "D": "-0.25"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "What is the value of (1/3)⁻¹? 🔄", "options": {"A": "-3", "B": "3", "C": "1/3", "D": "0"}, "correct_answer": "B"},
    {"id": 4, "difficulty": "easy", "question": "Express 4¹/² in simplest form. 🌱", "options": {"A": "2", "B": "√2", "C": "16", "D": "0.5"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "Find: 8¹/³ = ? ⬜", "options": {"A": "2", "B": "4", "C": "√8", "D": "24"}, "correct_answer": "A"},
    {"id": 6, "difficulty": "easy", "question": "What is 10⁻²? 📉", "options": {"A": "100", "B": "0.1", "C": "0.01", "D": "-100"}, "correct_answer": "C"},
    {"id": 7, "difficulty": "easy", "question": "Simplify: x⁻³ × x² = ? (x ≠ 0) 🤝", "options": {"A": "x⁻⁶", "B": "x⁻¹", "C": "x⁵", "D": "x⁶"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "Calculate: (9)¹/² = ? ☀️", "options": {"A": "4.5", "B": "81", "C": "3", "D": "18"}, "correct_answer": "C"},
    {"id": 9, "difficulty": "easy", "question": "What is the reciprocal of 7⁻¹? 🔁", "options": {"A": "7", "B": "1/7", "C": "-7", "D": "-1/7"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "Simplify: (25)⁻¹/² = ? ⭐", "options": {"A": "5", "B": "-5", "C": "1/5", "D": "0.2"}, "correct_answer": "C"},
    {"id": 11, "difficulty": "easy", "question": "Find: 27¹/³ = ? 🧊", "options": {"A": "9", "B": "√27", "C": "3", "D": "81"}, "correct_answer": "C"},
    {"id": 12, "difficulty": "easy", "question": "What is (1/4)⁻²? 🚀", "options": {"A": "-16", "B": "1/16", "C": "8", "D": "16"}, "correct_answer": "D"},
    {"id": 13, "difficulty": "easy", "question": "Simplify: a⁰ ÷ a⁻² = ? (a ≠ 0) 🎈", "options": {"A": "a²", "B": "a⁻²", "C": "1", "D": "0"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "Express 125⁻¹/³. 📏", "options": {"A": "5", "B": "1/5", "C": "-5", "D": "25"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "Solve: (16)¹/⁴ = ? 🔢", "options": {"A": "2", "B": "4", "C": "8", "D": "256"}, "correct_answer": "B"},
    {"id": 16, "difficulty": "easy", "question": "What is 2⁻³ + 2⁻²? ➕", "options": {"A": "3/8", "B": "5/8", "C": "1/8", "D": "1/4"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "Simplify: (1/5)⁻¹ × 5⁻¹ = ? ✖️", "options": {"A": "1", "B": "25", "C": "1/25", "D": "5"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "Find: (100)⁻¹/² = ? 🔟", "options": {"A": "10", "B": "0.1", "C": "-10", "D": "0.01"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "What is (x²)⁻¹? (x ≠ 0) 🌀", "options": {"A": "x⁻²", "B": "x²", "C": "x", "D": "1"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "Simplify: 3⁻¹ + 3⁰ = ? 🧮", "options": {"A": "4/3", "B": "1", "C": "2/3", "D": "3/2"}, "correct_answer": "A"},
    {"id": 21, "difficulty": "medium", "question": "Simplify: (64⁻²/³) = ? 🧱", "options": {"A": "1/16", "B": "16", "C": "-16", "D": "1/4"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "Find x: 2ˣ = 1/32. 🔎", "options": {"A": "5", "B": "-5", "C": "6", "D": "-6"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "Simplify: (8/27)⁻²/³ = ? ⚗️", "options": {"A": "4/9", "B": "9/4", "C": "27/8", "D": "-9/4"}, "correct_answer": "B"},
    {"id": 24, "difficulty": "medium", "question": "Express 2⁻³ × 4² as a power of 2. 🔄", "options": {"A": "2⁻¹", "B": "2¹", "C": "2⁻⁶", "D": "2⁵"}, "correct_answer": "B"},
    {"id": 25, "difficulty": "medium", "question": "Simplify: (x⁻¹y²) / (x²y⁻³) (x,y ≠ 0). 🍃", "options": {"A": "y⁵/x³", "B": "x³y⁵", "C": "x/y", "D": "y/x³"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Find the value of: (0.125)⁻¹/³. 💎", "options": {"A": "2", "B": "4", "C": "8", "D": "0.5"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "If 5⁻ᵐ = 1/125, find m. 🎯", "options": {"A": "3", "B": "-3", "C": "1/3", "D": "-1/3"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "Simplify: √(a⁶b⁻⁴) = ? (a,b > 0). 🌿", "options": {"A": "a³b⁻²", "B": "a³b²", "C": "a⁴b⁻²", "D": "a⁶b⁻²"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "Evaluate: (32)³/⁵. 📊", "options": {"A": "2", "B": "4", "C": "8", "D": "16"}, "correct_answer": "C"},
    {"id": 30, "difficulty": "medium", "question": "Solve for n: (1/9)ⁿ = 243. 🤔", "options": {"A": "-2.5", "B": "2.5", "C": "-5/2", "D": "5/2"}, "correct_answer": "C"},
    {"id": 31, "difficulty": "medium", "question": "Simplify: [(3⁻² × 4²)/(3 × 4⁻¹)]² = ? 🌀", "options": {"A": "64/81", "B": "81/64", "C": "16/9", "D": "9/16"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "Express (a¹/³ × a²/³) / a⁻¹/³ as a single power. ✨", "options": {"A": "a²", "B": "a⁴/³", "C": "a", "D": "a⁰"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "Find x if (√2)ˣ = 0.125. 🧭", "options": {"A": "-6", "B": "6", "C": "-3", "D": "3"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "Simplify: (x⁻² + y⁻²) / (x⁻¹y⁻¹) (x,y ≠ 0). 🤝", "options": {"A": "(x²+y²)/(xy)", "B": "xy/(x²+y²)", "C": "(x+y)/(xy)", "D": "(1/x²)+(1/y²)"}, "correct_answer": "B"},
    {"id": 35, "difficulty": "medium", "question": "Evaluate: 4³/² × 4⁻¹/² = ? ⚡", "options": {"A": "2", "B": "4", "C": "8", "D": "16"}, "correct_answer": "B"},
    {"id": 36, "difficulty": "medium", "question": "If (0.04)⁻¹/² = 5ⁿ, find n. 🔑", "options": {"A": "1", "B": "2", "C": "-1", "D": "-2"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "Simplify: (a¹/²b⁻¹/³)⁶ ÷ (a²b⁻¹) = ? 📦", "options": {"A": "ab", "B": "a", "C": "b", "D": "a²b"}, "correct_answer": "B"},
    {"id": 38, "difficulty": "medium", "question": "Find the reciprocal of 2⁻¹ + 3⁻¹. 🔁", "options": {"A": "5/6", "B": "6/5", "C": "1/5", "D": "5"}, "correct_answer": "B"},
    {"id": 39, "difficulty": "medium", "question": "Solve: 2ˣ⁻² = 8⁻ˣ. ⚖️", "options": {"A": "2", "B": "1", "C": "0", "D": "-1"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "Simplify: (9ⁿ⁺¹ × 3ⁿ⁻¹) / (27ⁿ) = ? 🧬", "options": {"A": "1", "B": "3", "C": "9", "D": "1/3"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "hard", "question": "Simplify: [ { (x⁻² y)⁻³ / (x y⁻²)² } ]⁻¹ = ? 🏆", "options": {"A": "x⁸ y⁻⁷", "B": "x⁴ y⁻¹", "C": "x⁻⁸ y⁷", "D": "x⁻¹⁶ y¹³"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "hard", "question": "If 2ˣ = 3ʸ = 6⁻ᶻ, find (1/x)+(1/y)+(1/z). 🤯", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "hard", "question": "Evaluate: (0.125)⁻²/³ × (0.25)⁻³/². 🌌", "options": {"A": "16", "B": "32", "C": "64", "D": "128"}, "correct_answer": "C"},
    {"id": 44, "difficulty": "hard", "question": "Simplify: (a¹/² + a⁻¹/²)² - (a¹/² - a⁻¹/²)² = ? 🧩", "options": {"A": "2", "B": "4", "C": "2a", "D": "4a⁰"}, "correct_answer": "B"},
    {"id": 45, "difficulty": "hard", "question": "Solve for x: 4ˣ⁻¹ = (√8)³⁻ˣ. ⚔️", "options": {"A": "11/7", "B": "13/7", "C": "15/7", "D": "17/7"}, "correct_answer": "B"},
    {"id": 46, "difficulty": "hard", "question": "If a = 2¹/² + 2⁻¹/² and b = 2¹/² - 2⁻¹/², find a² + b². 🛡️", "options": {"A": "3", "B": "4", "C": "6", "D": "8"}, "correct_answer": "C"},
    {"id": 47, "difficulty": "hard", "question": "Simplify: [ (xᵐ / xⁿ)ᵐ⁺ⁿ × (xⁿ / xᵖ)ⁿ⁺ᵖ × (xᵖ / xᵐ)ᵖ⁺ᵐ ]¹/². ♾️", "options": {"A": "0", "B": "1", "C": "x", "D": "xᵐⁿᵖ"}, "correct_answer": "B"},
    {"id": 48, "difficulty": "hard", "question": "Find the value of: (256⁰⋅¹⁶ × 256⁰⋅⁰⁹) = ? 🔢", "options": {"A": "4", "B": "16", "C": "64", "D": "256"}, "correct_answer": "B"},
    {"id": 49, "difficulty": "hard", "question": "If 5ˣ⁻ʸ = 125 and 5ˣ⁺ʸ = 1/25, find x and y. Then find 2x+y. 🎭", "options": {"A": "-1", "B": "0", "C": "1", "D": "2"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "hard", "question": "Simplify: (√a √b)⁻² × (a⁻¹/² b⁻¹/²)⁴ ÷ (a⁻² b⁻²). 🌀", "options": {"A": "a²b²", "B": "a⁴b⁴", "C": "1", "D": "(ab)²"}, "correct_answer": "C"},
    {"id": 51, "difficulty": "hard", "question": "Prove that: (xᵃ⁻ᵇ)ᵃ⁺ᵇ.(xᵇ⁻ᶜ)ᵇ⁺ᶜ.(xᶜ⁻ᵃ)ᶜ⁺ᵃ = 1. Find its value if a=1,b=2,c=3. ✅", "options": {"A": "0", "B": "1", "C": "x", "D": "x⁻⁶"}, "correct_answer": "B"},
    {"id": 52, "difficulty": "hard", "question": "Evaluate: [{(0.064)⁻²/³} / {(0.0625)⁻³/⁴}] = ? 🧮", "options": {"A": "5/4", "B": "4/5", "C": "25/16", "D": "16/25"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "hard", "question": "If 3²ˣ⁺¹ / 9ˣ⁺¹ = 27⁻ˣ, find x². 🎢", "options": {"A": "1", "B": "4", "C": "9", "D": "16"}, "correct_answer": "C"},
    {"id": 54, "difficulty": "hard", "question": "Simplify: (aⁿ⁺¹ × bⁿ⁻¹) / (aⁿ⁻¹ × bⁿ⁺¹) all to the power of 1/2. 🌱", "options": {"A": "a/b", "B": "b/a", "C": "√(ab)", "D": "1"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "hard", "question": "Solve: 2²ˣ - 6 × 2ˣ + 8 = 0. ⏳", "options": {"A": "1, 2", "B": "0, 3", "C": "1, 3", "D": "2, 3"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "hard", "question": "Find the simplest form of: (x¹/³ y⁻¹/²)¹² ÷ (x⁴ y⁻⁶). 🍀", "options": {"A": "1", "B": "x⁰y⁰", "C": "y⁶", "D": "x⁻⁴y⁰"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "hard", "question": "If (√5)⁷ ÷ (√5)⁵ = 5ⁿ, find n. ⚡", "options": {"A": "1", "B": "5", "C": "6", "D": "1/5"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "hard", "question": "Evaluate: (27/64)⁻²/³ + (25/36)⁻¹/². 🏹", "options": {"A": "100/9", "B": "9/100", "C": "241/144", "D": "144/241"}, "correct_answer": "C"},
    {"id": 59, "difficulty": "hard", "question": "Simplify: [(a⁻¹ + b⁻¹)/(a⁻² - b⁻²)]⁻¹. 🪐", "options": {"A": "(a-b)/ab", "B": "ab/(a-b)", "C": "(a+b)/ab", "D": "ab/(a+b)"}, "correct_answer": "B"},
    {"id": 60, "difficulty": "hard", "question": "If a = 2 - √3, find the value of a¹/² + a⁻¹/². 🌳", "options": {"A": "√6", "B": "2√2", "C": "√12", "D": "√3"}, "correct_answer": "A"}
  ]
}