Class 8 Notes

Class 8 Mathematics Notes Chapter 12 (Exponents and Powers) – Mathematics Book

Philoid

Philoid

— 5 min read

Mathematics
Alright class, let's get straight into Chapter 12: Exponents and Powers. This is a fundamental chapter, and mastering these concepts is crucial not just for your Class 8 exams, but also forms the base for many topics you'll encounter in higher mathematics and competitive exams. Pay close attention!

Chapter 12: Exponents and Powers - Detailed Notes

1. Introduction to Exponents and Powers

  • Power: An expression representing repeated multiplication of the same factor. It has two parts: a base and an exponent.
  • Base: The number or factor that is multiplied by itself.
  • Exponent (or Index or Power): The small number written above and to the right of the base. It indicates how many times the base is to be multiplied by itself.
    • Example: In 10⁴ (read as "10 raised to the power of 4" or "the fourth power of 10"),
      • Base = 10
      • Exponent = 4
      • Meaning = 10 × 10 × 10 × 10 = 10,000
    • Example: In (-2)³ (read as "negative 2 raised to the power of 3"),
      • Base = -2
      • Exponent = 3
      • Meaning = (-2) × (-2) × (-2) = -8

2. Negative Exponents

  • For any non-zero integer 'a', the expression a⁻ᵐ (where 'm' is a positive integer) is the multiplicative inverse or reciprocal of aᵐ.
  • Rule: a⁻ᵐ = 1 / aᵐ
    • Example: 2⁻³ = 1 / 2³ = 1 / (2 × 2 × 2) = 1/8
    • Example: 10⁻² = 1 / 10² = 1 / (10 × 10) = 1/100 = 0.01
  • Similarly, 1 / a⁻ᵐ = aᵐ
    • Example: 1 / 5⁻² = 5² = 25

3. Laws of Exponents

These laws are essential for simplifying expressions involving exponents. Let 'a' and 'b' be any non-zero integers, and 'm' and 'n' be any integers.

  • Law 1: Multiplying Powers with the Same Base

    • Rule: aᵐ × aⁿ = aᵐ⁺ⁿ
    • Explanation: When multiplying powers with the same base, keep the base and add the exponents.
    • Example: 2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128
    • Example: (-3)² × (-3)³ = (-3)²⁺³ = (-3)⁵ = -243

  • Law 2: Dividing Powers with the Same Base

    • Rule: aᵐ / aⁿ = aᵐ⁻ⁿ
    • Explanation: When dividing powers with the same base, keep the base and subtract the exponents (exponent of the numerator minus exponent of the denominator).
    • Example: 5⁶ / 5² = 5⁶⁻² = 5⁴ = 625
    • Example: 7³ / 7⁵ = 7³⁻⁵ = 7⁻² = 1 / 7² = 1/49

  • Law 3: Taking Power of a Power

    • Rule: (aᵐ)ⁿ = aᵐⁿ
    • Explanation: When raising a power to another power, keep the base and multiply the exponents.
    • Example: (2³)⁴ = 2³ˣ⁴ = 2¹² = 4096
    • Example: ((-1)²)⁵ = (-1)²ˣ⁵ = (-1)¹⁰ = 1 (Remember: -1 raised to an even power is 1)

  • Law 4: Multiplying Powers with the Same Exponent

    • Rule: aᵐ × bᵐ = (ab)ᵐ
    • Explanation: When multiplying powers with different bases but the same exponent, multiply the bases and keep the exponent.
    • Example: 2³ × 3³ = (2 × 3)³ = 6³ = 216
    • Example: (-2)⁴ × 5⁴ = (-2 × 5)⁴ = (-10)⁴ = 10,000

  • Law 5: Dividing Powers with the Same Exponent

    • Rule: aᵐ / bᵐ = (a/b)ᵐ
    • Explanation: When dividing powers with different bases but the same exponent, divide the bases and keep the exponent.
    • Example: 6⁵ / 3⁵ = (6/3)⁵ = 2⁵ = 32
    • Example: 4⁷ / 2⁷ = (4/2)⁷ = 2⁷ = 128

  • Law 6: Power with Exponent Zero

    • Rule: a⁰ = 1 (where a ≠ 0)
    • Explanation: Any non-zero number raised to the power of zero is equal to 1.
    • Example: 100⁰ = 1
    • Example: (-5)⁰ = 1
    • Example: (2/3)⁰ = 1

4. Use of Exponents to Express Numbers in Standard Form (Scientific Notation)

  • Purpose: To write very large or very small numbers concisely.

  • Standard Form: A number is expressed as k × 10ⁿ, where:

    • 'k' is a decimal number such that 1 ≤ k < 10.
    • 'n' is an integer (positive for large numbers, negative for small numbers).

  • Expressing Large Numbers in Standard Form:

    • Move the decimal point to the left until you get a number between 1 and 10 (i.e., just after the first non-zero digit).
    • The number of places the decimal point was moved gives the positive exponent 'n' for the power of 10.
    • Example: 345,000,000 = 3.45 × 10⁸ (Decimal moved 8 places left)
    • Example: 1500 = 1.5 × 10³ (Decimal moved 3 places left)

  • Expressing Small Numbers (less than 1) in Standard Form:

    • Move the decimal point to the right until you get a number between 1 and 10 (i.e., just after the first non-zero digit).
    • The number of places the decimal point was moved gives the negative exponent 'n' for the power of 10.
    • Example: 0.000056 = 5.6 × 10⁻⁵ (Decimal moved 5 places right)
    • Example: 0.0081 = 8.1 × 10⁻³ (Decimal moved 3 places right)

  • Converting from Standard Form to Usual Form:

    • If the exponent 'n' is positive, move the decimal point 'n' places to the right (adding zeros if needed).
    • If the exponent 'n' is negative, move the decimal point 'n' places to the left (adding zeros if needed).
    • Example: 2.7 × 10⁵ = 270,000
    • Example: 9.1 × 10⁻⁴ = 0.00091

Key Takeaways for Exams:

  • Memorize the Laws of Exponents thoroughly.
  • Practice applying these laws to simplify complex expressions. Pay attention to signs, especially with negative bases.
  • Understand the concept of negative exponents as reciprocals.
  • Master the conversion between usual form and standard form (scientific notation) for both large and small numbers.
  • Remember a⁰ = 1 for any non-zero 'a'.

Now, let's test your understanding with some multiple-choice questions.

Multiple Choice Questions (MCQs)

  1. The value of (3⁻²) is:
    a) 9
    b) -9
    c) 1/9
    d) 1/6

  2. The value of (-4)⁵ / (-4)³ is:
    a) (-4)⁸
    b) 16
    c) -16
    d) (-4)²

  3. The value of (2³)⁴ is:
    a) 2⁷
    b) 2¹²
    c) 2⁶
    d) 8⁴

  4. Which of the following is equal to (5/7)⁻³?
    a) (7/5)³
    b) (5/7)³
    c) (-5/7)³
    d) -(5/7)³

  5. The standard form of 0.000064 is:
    a) 6.4 × 10⁵
    b) 6.4 × 10⁻⁵
    c) 64 × 10⁻⁶
    d) 0.64 × 10⁻⁴

  6. The usual form of 3.05 × 10⁶ is:
    a) 305000
    b) 3050000
    c) 30.5 × 10⁵
    d) 0.00000305

  7. The value of (2⁰ + 3⁰ + 4⁰) is:
    a) 9
    b) 1
    c) 3
    d) 24

  8. The value of [(1/3)⁻² + (1/4)⁻²] is:
    a) 7
    b) 1/25
    c) 25
    d) 1/7

  9. If (2/5)³ × (2/5)⁻⁶ = (2/5)²ˣ⁻¹, then the value of x is:
    a) -1
    b) 1
    c) -2
    d) 2

  10. The value of (-1)¹⁰⁰ is:
    a) -1
    b) 1
    c) 100
    d) -100

Answer Key & Explanations

  1. c) 1/9 (Explanation: a⁻ᵐ = 1/aᵐ. So, 3⁻² = 1/3² = 1/9)
  2. b) 16 (Explanation: aᵐ / aⁿ = aᵐ⁻ⁿ. So, (-4)⁵ / (-4)³ = (-4)⁵⁻³ = (-4)² = 16)
  3. b) 2¹² (Explanation: (aᵐ)ⁿ = aᵐⁿ. So, (2³)⁴ = 2³ˣ⁴ = 2¹²)
  4. a) (7/5)³ (Explanation: (a/b)⁻ᵐ = (b/a)ᵐ. So, (5/7)⁻³ = (7/5)³)
  5. b) 6.4 × 10⁻⁵ (Explanation: Move decimal 5 places right: 0.00006.4 -> 6.4. Since moved right, exponent is negative: 6.4 × 10⁻⁵)
  6. b) 3050000 (Explanation: Exponent is +6. Move decimal 6 places right: 3.05 -> 3050000.)
  7. c) 3 (Explanation: a⁰ = 1. So, 2⁰ + 3⁰ + 4⁰ = 1 + 1 + 1 = 3)
  8. c) 25 (Explanation: (1/3)⁻² = 3² = 9. (1/4)⁻² = 4² = 16. Sum = 9 + 16 = 25)
  9. a) -1 (Explanation: (2/5)³⁺⁽⁻⁶⁾ = (2/5)²ˣ⁻¹. (2/5)⁻³ = (2/5)²ˣ⁻¹. Equating exponents: -3 = 2x - 1 => -2 = 2x => x = -1)
  10. b) 1 (Explanation: -1 raised to any even integer power is 1.)

Study these notes carefully and practice solving problems using the laws of exponents. Good luck with your preparation!

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