Overview
This lecture introduces the laws of indices (exponents), focusing on how to combine and simplify expressions with the same base using the rules of exponents.
Law of Indices: Basics
- Indices (exponents) represent repeated multiplication, e.g., ( m \times m = m^2 ).
- The product law states: ( a^m \times a^n = a^{m+n} ) for the same base.
Combining Powers
- Multiplying terms with the same base adds their exponents, e.g., ( n^2 \times n^3 = n^{2+3} = n^5 ).
- Expressions like ( y \times y \times y ) become ( y^3 ).
- When multiplying more than two terms: ( a^2 \times a^3 \times a = a^{2+3+1} = a^6 ).
Working with Brackets
- Powers inside brackets, e.g., ( (y^2)^3 ), follow: ( (a^m)^n = a^{mn} ).
- If multiplying a term with a bracketed power: ( y \times (y^2)^3 ), apply the rules step by step.
Practice Questions
- Simplify products like ( m \times m \times m ) and write as ( m^3 ).
- Combine terms such as ( n^2 \times n^4 ) into a single power.
- Express complex terms like ( (y^2)^3 ) as ( y^6 ).
- Multiply terms with coefficients: ( 2x \times 2x = (2 \times 2) \times (x \times x) = 4x^2 ).
Key Terms & Definitions
- Index (Exponent) — Indicates how many times a base is multiplied by itself.
- Base — The number or variable being multiplied repeatedly.
- Product Law of Indices — When multiplying like bases, add the exponents: ( a^m \times a^n = a^{m+n} ).
- Power of a Power — Raising a power to another power multiplies the exponents: ( (a^m)^n = a^{mn} ).
Action Items / Next Steps
- Complete all practice questions simplifying products of powers as single exponents.
- Review definitions and laws before the next session.
- Watch Video 17 on www.corbettmaths.com for further examples.