Short Notes: Power Play | Mathematics Class 8- New NCERT (Ganita Prakash) PDF Download

Exponential Notation

Definition: Shorthand for repeated multiplication of the same number.

Examples:

• n² = n × n

• n³ = n × n × n

• n⁴ = n × n × n × n

Algebra:

• a³ × b² = a × a × a × b × b

• a² × b⁴ = a × a × b × b × b × b

Important Note: Addition is not exponent:

• 4 + 4 + 4 = 3 × 4 = 12

• 4 × 4 × 4 = 4³ = 64

Prime Factorization and Exponential Form

Prime factorization is expressing a number as a product of its prime numbers.

Exponential form is writing repeated prime factors using powers.

Steps to Express in Exponential Form

1. Find the prime factors of the number.

2. Group the same factors together.

3. Write each group as a power.

4. Combine to get the exponential form.

Example: Express 32,400 in exponential form

1. Prime Factorization:
32,400 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5

2. Group factors:

• 2 × 2 × 2 × 2 = 2⁴

• 3 × 3 × 3 × 3 = 3⁴

• 5 × 5 = 5²

3. Exponential Form:
32,400 = 2⁴ × 3⁴ × 5²

Quick Tip: Prime factorization is useful for finding HCF, LCM, and simplifying roots of numbers.

Laws of Exponents: Multiplication and Division of Powers

1. Multiplying Same Bases

• Rule: Add the exponents

• Formula: nᵃ × nᵇ = nᵃ⁺ᵇ

• Example: p⁴ × p⁶ = p¹⁰

2. Power of a Power

• Rule: Multiply the exponents

• Formula: (nᵃ)ᵇ = nᵃ×ᵇ

• Example: (2⁵)² = 2¹⁰

3. Dividing Powers with Same Base

• Rule: Subtract the exponents (denominator from numerator)

• Formula: nᵃ ÷ nᵇ = nᵃ⁻ᵇ, n ≠ 0

• Example: 2⁴ ÷ 2³ = 2¹

4. Negative Powers

• Rule: Reciprocal of the positive power

• Formula: n⁻ᵃ = 1 ÷ nᵃ

• Example: 3⁻² = 1 ÷ 3² = 1/9

5. Zero Exponent

• Rule: Any non-zero number to the power 0 is 1

• Formula: x⁰ = 1, x ≠ 0

• Example: 7⁰ = 1

6. Multiplying Different Bases with Same Exponent

• Rule: Multiply the bases, keep the exponent

• Formula: mᵃ × nᵃ = (m × n)ᵃ

• Example: 2³ × 5³ = (2 × 5)³ = 10³

7. Dividing Different Bases with Same Exponent

• Rule: Divide the bases, keep the exponent

• Formula: mᵃ ÷ nᵃ = (m ÷ n)ᵃ

• Example: 8² ÷ 2² = (8 ÷ 2)² = 4²

 Linear vs. Exponential Growth

1. Linear Growth

Description: Adds a fixed amount per step.

Example:

• Distance to the Moon: 384,400 km = 384,400,000 m

• Step size: 20 cm = 0.2 m

• Number of steps:
  384,400,000 / 0.2 = 1,922,000,000 steps = 1.922 × 10⁹

2. Exponential Growth

Description: Multiplies by a fixed factor per step.

Example: Paper folding to the Moon:

• Initial thickness: 0.001 cm

• Number of folds: 46

• Thickness:
  T = 0.001 × 2⁴⁶ ≈ 7,036,874,841,600 cm ≈ 703,687.48 km

Powers of 10 and Scientific Notation

1. Expanded Form Using Powers of 10

For Whole Numbers:
Example: 47,561

• Expanded form:
  4 × 10⁴ + 7 × 10³ + 5 × 10² + 6 × 10¹ + 1 × 10⁰

For Decimals:
Example: 561.903

• Expanded form:
  5 × 10² + 6 × 10¹ + 1 × 10⁰ + 9 × 10⁻¹ + 0 × 10⁻² + 3 × 10⁻³

2. Scientific Notation

Any number can be written as: x × 10ʸ, where:

• x = coefficient (usually between 1 and 10)

• y = exponent (shows the scale of the number)

Examples:

• 5,900 → 5.9 × 10³

• 8,000,000 → 8 × 10⁶

Importance of Exponent: Determines scale; coefficient adjusts precision.

Importance of the Exponent

• Exponent (y) determines the scale or magnitude of the number.

• Coefficient (x) adjusts precision for significant digits.

Large Numbers in Nature

• Human population (2025) = 8 × 10⁹

• African elephants = 4 × 10⁵ → ~20,000 people per elephant

• Grains of sand on Earth ≈ 10²¹

• Stars in observable universe ≈ 2 × 10²³

• Drops of water on Earth ≈ 2 × 10²⁵

Fun Fact:

• 10⁶ seconds ≈ 11.6 days

• 10⁹ seconds ≈ 31.7 years