Introduction to Laws of Indices

Study Notes

Introduction to Laws of Indices

Indices (or exponents) represent repeated multiplication of a base. For example, 𝑥³ means 𝑥 × 𝑥 × 𝑥.
Understanding the laws of indices is crucial for simplifying and manipulating algebraic expressions involving powers.

Basic Laws

Law 1: Product of Powers
- When multiplying terms with the same base, add the exponents. 𝑥^𝑎 × 𝑥^𝑏 = 𝑥^𝑎+𝑏
Law 2: Quotient of Powers
- When dividing terms with the same base, subtract the exponents. 𝑥^𝑎 ÷ 𝑥^𝑏 = 𝑥^𝑎−𝑏
Law 3: Power of a Power
- To raise a power to another power, multiply the exponents. (𝑥^𝑎)^𝑏 = 𝑥^𝑎𝑏

Further Laws

Law 4: Power of a Product
- Raising a product to a power is equivalent to raising each factor to that power. (𝑥𝑦)^𝑛 = 𝑥^𝑛𝑦^𝑛
Law 5: Power of a Quotient
- Raising a quotient to a power is equivalent to raising both the numerator and denominator to that power. (𝑥/𝑦)^𝑛 = 𝑥^𝑛/𝑦^𝑛
Law 6: Zero Index
- Any non-zero number raised to the power of zero is equal to one 𝑥⁰ = 1 (where 𝑥 ≠ 0)
Law 7: Negative Indices
- A term with a negative index can be rewritten as its reciprocal with a positive index. 𝑥^−𝑛 = 1/𝑥^𝑛

Applications and Examples

Simplifying expressions involving indices, such as 𝑥³𝑦² × 𝑥⁻²𝑦⁴
Example: Simplify (2𝑥²𝑦)³ - Applying Law 4: 2³ × (𝑥²)³ × 𝑦³ - Applying Law 3: 8 × 𝑥⁶ × 𝑦³ = 8𝑥⁶𝑦³
Solving equations involving indices, like 2^𝑥 = 8. Logarithms are often used for such equations.
Example: Solve 𝑥² = 4 - Taking the square root of both sides, we get 𝑥 = ±2.
Calculating compound interest or exponential growth. The laws of indices provide a framework to manipulate these mathematical models.

Special Cases

Law 8: Fractional Indices
- Fractional indices represent roots. 𝑥^1/𝑛 = √𝑛𝑥
Example: 𝑥^1/2 = √𝑥; and 𝑥^2/3 = √𝑥²
Law 9: Scientific Notation
- Expressing very large or very small numbers in a standard form (e.g., 2.5 x 10³).

Important Considerations

The base must be the same for the product and quotient laws to apply.
Carefully consider the order of operations when evaluating expressions involving indices.
Fractional indices relate to roots and radicals.
Positive and negative exponents have specific interpretations in terms of scaling.

Description

This quiz covers the fundamental laws of indices, including the product of powers, quotient of powers, and power of a power. Understanding these laws is essential for simplifying algebraic expressions involving exponents. Test your knowledge and grasp these basic concepts in exponentiation.