Vardaan
Class 7 Maths • Chapter 13

Exponents and Powers

Vardaan Learning Institute • Detailed Notes with Practice Questions

⚡ 1. What is an Exponent?

aⁿ = a × a × a × ... (n times)
Here: a = base, n = exponent/power
Read as: "a raised to the power n" or "a to the nth power"

Examples:
• 2⁵ = 2×2×2×2×2 = 32
• (−3)⁴ = (−3)×(−3)×(−3)×(−3) = 81 (positive — even power)
• (−2)³ = (−2)×(−2)×(−2) = −8 (negative — odd power)
• (2/3)² = 4/9
🧠 Sign Rule for Negative Bases • Even power → positive result: (−a)ⁿ = positive if n is even
• Odd power → negative result: (−a)ⁿ = negative if n is odd

📋 2. Laws of Exponents

Law 1 — Product
aᵐ × aⁿ = aᵐ⁺ⁿ
2⁴ × 2³ = 2⁷ = 128
Law 2 — Quotient
aᵐ ÷ aⁿ = aᵐ⁻ⁿ (m>n)
3⁷ ÷ 3⁴ = 3³ = 27
Law 3 — Power of Power
(aᵐ)ⁿ = aᵐⁿ
(2³)⁴ = 2¹² = 4096
Law 4 — Product of Powers
(ab)ⁿ = aⁿ × bⁿ
(2×3)³ = 2³ × 3³ = 8×27=216
Law 5 — Quotient of Powers
(a/b)ⁿ = aⁿ/bⁿ
(2/5)³ = 8/125
Law 6 — Zero Exponent
a⁰ = 1 (a ≠ 0)
7⁰ = 1, (−5)⁰ = 1, 100⁰ = 1

🔢 3. Powers of 10 — Expressing Large Numbers

Standard Form Expanded Form Number
10¹ 10 Ten
10² 100 Hundred
10³ 1,000 Thousand
10⁶ 1,000,000 Million
10⁷ 10,000,000 One Crore
10⁹ 1,000,000,000 Billion

🌍 4. Standard Form (Scientific Notation)

Large numbers can be expressed as a product of a number between 1 and 10 and a power of 10.

Standard Form = m × 10ⁿ where 1 ≤ m < 10

Examples:
• 7,00,000 = 7 × 10⁵
• 342,000,000 = 3.42 × 10⁸
• Sun-Earth distance: 1.5 × 10⁸ km
• Diameter of hydrogen atom: 1.06 × 10⁻¹⁰ m (note: negative powers for very small numbers — in higher classes)
Prime Factorisation with Exponents:
Example: Express 729 as a power.
729 = 3 × 3 × 3 × 3 × 3 × 3 = 3⁶

Example: 67500 = 2² × 3³ × 5⁴ (find by prime factorisation)
✏️ Practice Questions — Exponents and Powers (30 Questions)

Section A — Basics Easy

Q1. Find: 2⁶
Q2. Find: (−3)⁴ and (−3)⁵
Q3. Find: (2/3)³
Q4. Simplify: 3⁴ × 3² using laws
Q5. Simplify: 5⁷ ÷ 5⁴
Q6. Find: (4²)³
Q7. Simplify: (2 × 5)³
Q8. Find: 10⁰ × 5⁰ × 3⁰

Section B — Laws of Exponents Medium

Q9. Find x: 2ˣ = 128
Q10. Find x: 5ˣ⁺¹ = 625
Q11. Simplify: (3⁴ × 3⁵) ÷ 3⁶
Q12. Simplify: [(2³)²] ÷ 2⁴
Q13. Simplify: (2² × 3²) × (2³ × 3)
Q14. Simplify: (a³ × a⁵) ÷ a⁴
Q15. Express 1024 as a power of 2
Q16. Express 243 as a power of 3

Section C — Standard Form / Large Numbers Medium

Q17. Express in standard form: 42,000,000
Q18. Express in standard form: 750,00,000
Q19. Write as ordinary number: 3.6 × 10⁵
Q20. Write as ordinary number: 7.02 × 10⁷
Q21. Express using prime factorisation with exponents: 180
Q22. Express using prime factorisation: 1080

Section D — Challenge Hard

  1. Q23. Find x: 3˟ × 3⁴ = 3⁷ × 3²
  2. Q24. Simplify: [2⁴ × (3⁵ ÷ 3⁴)] × (2²)³
  3. Q25. If 2ⁿ = 32, find 2ⁿ⁺² and 2ⁿ⁻¹
  4. Q26. Evaluate: [2⁷ × 3⁴] ÷ [(2⁴) × (3²)]
  5. Q27. Simplify: (2/3)³ × (3/4)² × (4/2)¹
  6. Q28. The size of a red blood cell is 7×10⁻⁶ m and of a plant cell is 1.4×10⁻⁵ m. How many red cells equal one plant cell?
  7. Q29. Simplify: (2ˣ × 3ˣ⁺¹) ÷ (6ˣ⁻¹) [express in simplest form]
  8. Q30. Express the product of the first 5 prime numbers using prime factorisation with exponents.
✅ Key Answers: Q1:64 | Q2:81, -243 | Q3:8/27 | Q4:3⁶=729 | Q5:5³=125 | Q6:4⁶=4096 | Q8:1 | Q9:x=7 | Q10:x=3 | Q11:3³=27 | Q12:2⁴=16 | Q15:2¹⁰ | Q16:3⁵ | Q17:4.2×10⁷ | Q19:360000 | Q21:2²×3²×5 | Q25:2ⁿ⁺²=128, 2ⁿ⁻¹=16 | Q28:2 red cells

📝 Quick Revision

  1. aⁿ = a multiplied n times. 'a' is base, 'n' is exponent/power
  2. Negative base: Even power → positive. Odd power → negative
  3. Law 1: aᵐ × aⁿ = aᵐ⁺ⁿ | Law 2: aᵐ ÷ aⁿ = aᵐ⁻ⁿ | Law 3: (aᵐ)ⁿ = aᵐⁿ
  4. Law 4: (ab)ⁿ = aⁿbⁿ | Law 5: (a/b)ⁿ = aⁿ/bⁿ | Law 6: a⁰ = 1
  5. Standard form: m × 10ⁿ where 1 ≤ m < 10
  6. Large numbers → use powers of 10. Compress with scientific notation.