Overview
This lecture introduces the mathematics of indices (also known as exponents or powers), explains key rules for manipulating them, and demonstrates their application through examples.
Introduction to Indices
- Indices is the plural of "index," also called exponent or power.
- Indices represent how many times a base number is multiplied by itself.
- Example: (2^3 = 2 \times 2 \times 2).
Rules of Indices
- Multiplication Rule: (a^m \times a^n = a^{m+n}), valid for same bases.
- Division Rule: (a^m / a^n = a^{m-n}), valid when bases are the same.
- Power of a Power Rule: ((a^m)^n = a^{m \times n}).
- Zero Power Rule: (a^0 = 1) for any nonzero (a).
- Negative Power Rule: (a^{-n} = 1/a^n).
- Fractional Power Rule: (a^{1/n} = \sqrt[n]{a}); (a^{m/n} = (\sqrt[n]{a})^m).
- Product of Powers Rule: ((ab)^m = a^m b^m).
- Quotient of Powers Rule: ((a/b)^m = a^m / b^m).
- Powers do not distribute over addition or subtraction: ((a+b)^m \neq a^m + b^m).
- Equality of Powers: If (a^m = a^n) and (a \neq 0), then (m = n).
Indices with Negative Bases
- ((-a)^m = a^m) if (m) is even; ((-a)^m = -a^m) if (m) is odd.
Worked Examples
- Simplifying expressions using indices rules, e.g., converting decimal and fractional powers.
- Changing negative or fractional indices to positive forms.
- Breaking down complex expressions into products and quotients with positive indices.
- Applying rules to solve problems without using logarithm or mathematical tables.
Key Terms & Definitions
- Base — The main number being multiplied.
- Exponent/Index/Power — The number indicating how many times the base is multiplied by itself.
- Root — The inverse operation of raising to a power, often shown as fractional indices.
Action Items / Next Steps
- Try the assigned practice problem (part b in the final example) and post your solution in the comments.
- Review the seven key rules of indices.
- Practice converting negative and fractional powers to positive forms.
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