EXPONENTS & POWER

THE SECRETS OF EXPONENTS & POWERS

Introduction

Exponents and powers help us express very large or very small numbers in a compact and convenient way. Instead of writing 1000000, we can write it as 10⁶. Similarly, fractions like 1/1000 can be written as 10⁻³. Understanding exponents is essential in mathematics, science, and real-life applications like calculating population growth, measuring distances in space, or dealing with computer storage units.

1. What is an Exponent?

An exponent refers to the number of times a base number is multiplied by itself.

Example:

• 2³ = 2 × 2 × 2 = 8
• 5⁴ = 5 × 5 × 5 × 5 = 625

Smart Trick to Remember:

👉 Think of an exponent as a “power boost” to a number! Higher exponent = More power!

2. Laws of Exponents

Smart Trick to Remember:

📌 Zero Power Rule: Any number raised to power 0 is always 1. (e.g., 100⁰ = 1)
📌 Negative Power Rule: Flip the number! a⁻ⁿ = 1/aⁿ (e.g., 10⁻² = 1/10² = 1/100)

3. Expressing Large and Small Numbers with Exponents (Scientific Notation)

For Large Numbers:

Instead of writing 1000000000, write it as 10⁹.

For Small Numbers:

Instead of writing 0.00001, write it as 10⁻⁵.

Examples:

• 1,000,000 = 10⁶
• 0.0001 = 10⁻⁴

Smart Trick to Remember:

🔢 Count the number of zeros to determine the exponent!

• 1,000,000 → 6 zeros → 10⁶
• 0.0001 → 4 decimal places → 10⁻⁴

4. Applications of Exponents in Real Life

✅ Astronomy: Distance between stars (e.g., Sun to Earth = 1.5 × 10⁸ km)
✅ Computer Science: Storage sizes (e.g., 1 Gigabyte = 10⁹ bytes)
✅ Biology: Bacteria population growth
✅ Physics: Speed of light (3 × 10⁸ m/s)

Smart Trick to Remember:

🌟 “Exponents make life easier!” Instead of writing huge numbers, just use powers of 10.

5. Common Mistakes to Avoid

❌ 2³ + 2² ≠ 2⁵ (Wrong!) → 8 + 4 = 12 (Correct Calculation!)
❌ a⁻² ≠ -a² (Wrong!) → a⁻² = 1/a² (Correct Calculation!)
❌ 10⁰ ≠ 0 (Wrong!) → 10⁰ = 1 (Always true!)

🧠 EXPONENTS & POWERS – 50 MCQs (Moderate Level)

🔹 Questions (1–50)

📘 Concept & Basics (1–10)

1. What is the value of 252^525?
   A) 16
   B) 32
   C) 25
   D) 10
2. In 343^434, 3 is called:
   A) Exponent
   B) Base
   C) Power
   D) Product
3. In 535^353, 3 represents:
   A) Base
   B) Value
   C) Exponent
   D) Product
4. Which is equal to 10310^3103?
   A) 100
   B) 1000
   C) 10000
   D) 10
5. 727^272 equals:
   A) 14
   B) 21
   C) 49
   D) 28
6. a1=?a^1 = ?a1=?
   A) 0
   B) 1
   C) a
   D) a²
7. Value of 909^090:
   A) 0
   B) 9
   C) 1
   D) Undefined
8. 23×22=?2^3 \times 2^2 = ?23×22=?
   A) 2⁵
   B) 2⁶
   C) 2⁴
   D) 4
9. 54÷52=?5^4 ÷ 5^2 = ?54÷52=?
   A) 5²
   B) 5⁶
   C) 5³
   D) 5¹
10. (23)2=?(2^3)^2 = ?(23)2=?
    A) 2⁵
    B) 2⁶
    C) 2⁹
    D) 2³

📘 Laws of Exponents (11–25)

11. a3×a4=?a^3 \times a^4 = ?a3×a4=?
    A) a⁷
    B) a¹²
    C) a¹
    D) a⁶
12. x6÷x2=?x^6 ÷ x^2 = ?x6÷x2=?
    A) x³
    B) x⁴
    C) x⁸
    D) x²
13. (a2)3=?(a^2)^3 = ?(a2)3=?
    A) a⁵
    B) a⁶
    C) a⁸
    D) a⁹
14. 102×103=?10^2 \times 10^3 = ?102×103=?
    A) 10⁵
    B) 10⁶
    C) 10⁴
    D) 10²
15. 80=?8^0 = ?80=?
    A) 0
    B) 8
    C) 1
    D) Undefined
16. a−3=?a^{-3} = ?a−3=?
    A) -a³
    B) 1/a³
    C) a³
    D) 0
17. 2−2=?2^{-2} = ?2−2=?
    A) -4
    B) 1/4
    C) 4
    D) 1/2
18. x5×x−2=?x^5 \times x^{-2} = ?x5×x−2=?
    A) x³
    B) x⁷
    C) x⁻³
    D) x²
19. a7÷a7=?a^7 ÷ a^7 = ?a7÷a7=?
    A) a
    B) 0
    C) 1
    D) 7
20. (32)3=?(3^2)^3 = ?(32)3=?
    A) 3⁵
    B) 3⁶
    C) 3⁸
    D) 9³
21. 53×5−1=?5^3 × 5^{-1} = ?53×5−1=?
    A) 5²
    B) 5⁴
    C) 5³
    D) 5⁻²
22. x0+x1=?x^0 + x^1 = ?x0+x1=?
    A) x
    B) x + 1
    C) 1 + x
    D) Both B & C
23. 23+22=?2^3 + 2^2 = ?23+22=?
    A) 2⁵
    B) 12
    C) 10
    D) 8
24. 104÷102=?10^4 ÷ 10^2 = ?104÷102=?
    A) 10²
    B) 10⁶
    C) 10³
    D) 10¹
25. (x3×x2)2=?(x^3 × x^2)^2 = ?(x3×x2)2=?
    A) x⁵
    B) x¹⁰
    C) x⁷
    D) x⁶

📘 Scientific Notation (26–35)

26. $1,000,000=?$
    A) 10⁵
    B) 10⁶
    C) 10⁷
    D) 10⁴
27. $0.0001=?$
    A) 10⁻³
    B) 10⁻⁴
    C) 10⁻⁵
    D) 10⁻²
28. 5×103=?5 × 10^3 = ?5×103=?
    A) 500
    B) 5000
    C) 50
    D) 50000
29. 3×10−2=?3 × 10^{-2} = ?3×10−2=?
    A) 0.03
    B) 0.003
    C) 0.3
    D) 3
30. Which is scientific notation?
    A) 3000
    B) 3 × 10³
    C) 30 × 10²
    D) 0.003
31. 7×102=?7 × 10^2 = ?7×102=?
    A) 700
    B) 70
    C) 7000
    D) 7
32. 4×10−3=?4 × 10^{-3} = ?4×10−3=?
    A) 0.004
    B) 0.04
    C) 0.0004
    D) 4
33. 9×105=?9 × 10^5 = ?9×105=?
    A) 90000
    B) 900000
    C) 9000
    D) 90
34. Which is smallest?
    A) 10³
    B) 10²
    C) 10⁻³
    D) 10⁻²
35. 6×10−1=?6 × 10^{-1} = ?6×10−1=?
    A) 0.6
    B) 0.06
    C) 6
    D) 60

📘 Application & Mixed (36–50)

36. Speed of light is approx:
    A) 3×1053 × 10^53×105
    B) 3×1083 × 10^83×108
    C) 3×1023 × 10^23×102
    D) 3×1063 × 10^63×106
37. 1 Gigabyte =
    A) 10310^3103 bytes
    B) 10610^6106 bytes
    C) 10910^9109 bytes
    D) 101210^{12}1012 bytes
38. 1002=?100^2 = ?1002=?
    A) 1000
    B) 10000
    C) 100
    D) 100000
39. 10−2=?10^{-2} = ?10−2=?
    A) 100
    B) 1/100
    C) 0.1
    D) 10
40. 43=?4^3 = ?43=?
    A) 12
    B) 64
    C) 16
    D) 32
41. (2×103)(3×102)=?(2 × 10^3)(3 × 10^2) = ?(2×103)(3×102)=?
    A) 6×1056 × 10^56×105
    B) 5×1055 × 10^55×105
    C) 6×1066 × 10^66×106
    D) 6×1046 × 10^46×104
42. x2×x3×x4=?x^2 × x^3 × x^4 = ?x2×x3×x4=?
    A) x⁹
    B) x⁶
    C) x⁸
    D) x¹⁰
43. a−1×a2=?a^{-1} × a^2 = ?a−1×a2=?
    A) a
    B) a³
    C) 1
    D) a⁻¹
44. 52+53=?5^2 + 5^3 = ?52+53=?
    A) 5⁵
    B) 150
    C) 125
    D) 30
45. (102)0=?(10^2)^0 = ?(102)0=?
    A) 0
    B) 1
    C) 100
    D) Undefined
46. 32×42=?3^2 × 4^2 = ?32×42=?
    A) 12²
    B) 144
    C) 24
    D) 48
47. 24÷22=?2^4 ÷ 2^2 = ?24÷22=?
    A) 2²
    B) 2³
    C) 2⁴
    D) 2
48. 103×10−1=?10^3 × 10^{-1} = ?103×10−1=?
    A) 10²
    B) 10⁴
    C) 10¹
    D) 10³
49. (x2)0=?(x^2)^0 = ?(x2)0=?
    A) x²
    B) 1
    C) 0
    D) x
50. a5÷a2×a3=?a^5 ÷ a^2 × a^3 = ?a5÷a2×a3=?
    A) a⁶
    B) a⁵
    C) a⁴
    D) a³

✅ ANSWER KEY

1-B, 2-B, 3-C, 4-B, 5-C,
6-C, 7-C, 8-A, 9-A, 10-B,
11-A, 12-B, 13-B, 14-A, 15-C,
16-B, 17-B, 18-A, 19-C, 20-B,
21-A, 22-D, 23-B, 24-A, 25-B,
26-B, 27-B, 28-B, 29-A, 30-B,
31-A, 32-A, 33-B, 34-C, 35-A,
36-B, 37-C, 38-B, 39-B, 40-B,
41-A, 42-A, 43-A, 44-B, 45-B,
46-B, 47-A, 48-A, 49-B, 50-A