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

1. Exponential Notation

Formula: A number n multiplied by itself a times is expressed as: nᵃ

Example:
2 × 2 × 2 × 2 = 2⁴ = 16

2. Laws of Exponents

Here are the laws of exponents when a and b are non-zero integers and m, n are any integers.

a. Product Rule

Formula:
nᵃ × nᵇ = nᵃ⁺ᵇ

Example:
2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128

b. Power of a Power Rule

Formula:
(nᵃ)ᵇ = nᵃ×ᵇ = (nᵇ)ᵃ

Example:
(2³)² = 2³×² = 2⁶ = 64

c. Quotient Rule

Formula:
nᵃ ÷ nᵇ = nᵃ⁻ᵇ

Example:
2⁵ ÷ 2² = 2⁵⁻² = 2³ = 8

d. Product of Different Bases 

Formula:
nᵃ × mᵃ = (n × m)ᵃ

Example:
2³ × 3³ = (2 × 3)³ = 6³ = 216

e. Zero Exponent Rule

Formula:
n⁰ = 1 (n ≠ 0)

Example:
5⁰ = 1

f. Negative Exponent Rule

Formula:
n⁻ᵃ = 1/nᵃ

Example:
2⁻³ = 1/2³ = 1/8

3. Scientific Notation

Formula: A number is written as:
x × 10ᵃ where 1 ≤ x < 10, a is an integer

Example:

• Number: 59,853

• Scientific notation:
  59,853 = 5.9853 × 10⁴

4. Prime Factorization in Exponential Form

Formula: Express a number as a product of prime factors in exponential form.

Example:

• Number: 32,400

• Prime factorization:
  32,400 = 2⁴ × 3⁴ × 5²

5. Combinations

Formula: For k items, each with n choices, the total number of combinations is: nᵏ

Example:

A 5-digit password using digits 0–9:
10⁵ = 100,000 combinations

6. Linear vs. Exponential Growth

a. 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

b. 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