Introduction to Laws of Indices

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.

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.