Indices & Standard Form
Mastering powers and scientific notation for very large and small numbers.
### Introduction to Indices
Indices (also known as powers or exponents) are a fundamental concept in mathematics used to represent repeated multiplication. In the expression aⁿ, 'a' is called the base, and 'n' is the index or exponent. It tells us to multiply the base 'a' by itself 'n' times. For example, 3⁴ means 3 × 3 × 3 × 3 = 81.
### The Laws of Indices
To work efficiently with indices, we use a set of rules known as the laws of indices. These laws are essential for simplifying complex expressions and are frequently tested in Cambridge O Level examinations.
Formula: aᵐ × aⁿ = aᵐ⁺ⁿ
*Example*: 5³ × 5² = 5³⁺² = 5⁵ = 3125.
Formula: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
*Example*: 7⁶ ÷ 7⁴ = 7⁶⁻⁴ = 7² = 49.
Formula: (aᵐ)ⁿ = aᵐⁿ
*Example*: (2³)⁴ = 2³ˣ⁴ = 2¹² = 4096.
Formula: a⁰ = 1 (where a ≠ 0)
*Example*: 15⁰ = 1. This can be understood from the division law: a²/a² = a²⁻² = a⁰. Since any number divided by itself is 1, a⁰ must be 1.
Formula: a⁻ⁿ = 1/aⁿ
*Example*: 4⁻² = 1/4² = 1/16.
Formula: a¹/ⁿ = ⁿ√a
*Example*: 64¹/³ = ³√64 = 4 (since 4 × 4 × 4 = 64).
Formula: aᵐ/ⁿ = (ⁿ√a)ᵐ = ⁿ√(aᵐ)
*Example*: 8²/³ = (³√8)² = 2² = 4.
### Standard Form (Scientific Notation)
Standard form is a convenient way to write very large or very small numbers. It is used extensively in science and engineering. A number is in standard form when it is written as A × 10ⁿ, where:
Converting to Standard Form:
*Example*: The number 5,800,000 becomes 5.8 × 10⁶ (decimal moved 6 places left).
*Example*: The number 0.00045 becomes 4.5 × 10⁻⁴ (decimal moved 4 places right).
Converting from Standard Form:
*Example*: 3.91 × 10⁵ = 391,000.
*Example*: 8.2 × 10⁻³ = 0.0082.
Calculations with Standard Form:
To multiply or divide numbers in standard form, handle the 'A' parts and the '10ⁿ' parts separately, using the laws of indices for the powers of 10.
*Example*: (3 × 10⁵) × (2 × 10³) = (3 × 2) × (10⁵ × 10³) = 6 × 10⁸.
Key Points to Remember
- 1Multiplication Law: When bases are the same, add the indices (aᵐ × aⁿ = aᵐ⁺ⁿ).
- 2Division Law: When bases are the same, subtract the indices (aᵐ ÷ aⁿ = aᵐ⁻ⁿ).
- 3Power of a Power Law: To raise a power to another power, multiply the indices ((aᵐ)ⁿ = aᵐⁿ).
- 4Negative Index: A negative index signifies a reciprocal (a⁻ⁿ = 1/aⁿ).
- 5Fractional Index: This represents a root and a power (aᵐ/ⁿ = (ⁿ√a)ᵐ).
- 6Zero Index: Any non-zero base raised to the power of zero equals 1 (a⁰ = 1).
- 7Standard form is written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.
- 8Positive 'n' in standard form indicates a large number; negative 'n' indicates a small number.
Pakistan Example
Pakistan's Economy in Standard Form
In fiscal year 2023, the Gross Domestic Product (GDP) of Pakistan was approximately 84 trillion Pakistani Rupees (PKR). To express this massive number in standard form, we write it out as 84,000,000,000,000. We then move the decimal point from the end to a position after the first non-zero digit (8), which is 13 places to the left. Therefore, Pakistan's GDP can be written concisely as **8.4 × 10¹³ PKR**.
Quick Revision Infographic
Mathematics — Quick Revision
Indices & Standard Form
Key Concepts
Formulas to Know
Law: When bases are the same, add the indices (aᵐ × aⁿ = aᵐ⁺ⁿ).Law: When bases are the same, subtract the indices (aᵐ ÷ aⁿ = aᵐ⁻ⁿ).Power Law: To raise a power to another power, multiply the indices ((aᵐ)ⁿ = aᵐⁿ).Index: A negative index signifies a reciprocal (a⁻ⁿ = 1/aⁿ).Pakistan's Economy in Standard Form
In fiscal year 2023, the Gross Domestic Product (GDP) of Pakistan was approximately 84 trillion Pakistani Rupees (PKR). To express this massive number in standard form, we write it out as 84,000,000,000,000. We then move the decimal point from the end to a position after the first non-zero digit (8), which is 13 places to the left. Therefore, Pakistan's GDP can be written concisely as **8.4 × 10¹³ PKR**.