Overview
This lecture explains how to multiply and divide numbers with exponents (powers) when they have the same base, focusing on shortcut rules.
Multiplying Exponents with the Same Base
- Multiplication rules work only if terms have the same base.
- When multiplying, add the exponents: ( b^3 \times b^2 = b^{3+2} = b^5 ).
- The shortcut: ( x^m \times x^n = x^{m+n} ).
Dividing Exponents with the Same Base
- For division, subtract the exponents: ( a^6 \div a^3 = a^{6-3} = a^3 ).
- General rule: ( x^m \div x^n = x^{m-n} ).
Examples and Special Cases
- ( x^7 \times x^4 = x^{11} ) since ( 7+4=11 ).
- ( 2^7 \div 2^4 = 2^{3} = 8 ) (when the base is a number, you can compute the value).
- ( a^9 \times a^{-5} = a^{4} ) by adding ( 9+(-5) ).
- ( b^3 \div b^8 = b^{-5} ); negative exponents are valid.
- For three terms: ( a^2 \times a^4 \times a^{-3} = a^{3} ) (add all exponents).
- ( p^3 \div p = p^{3-1} = p^2 ); a lone variable has an unwritten exponent of 1.
Numbers and Letters Together
- Multiply numbers and apply exponent rules separately for variables.
- ( 3a^5 \times 4a^3 = 12a^8 ); multiply 3 by 4 and add exponents 5+3.
- ( 8b^{11} \div 2b^5 = 4b^6 ); divide 8 by 2 and subtract exponents 11-5.
Key Terms & Definitions
- Base — The number or letter being multiplied by itself.
- Exponent (Power) — Tells how many times to multiply the base by itself.
- Negative Exponent — Shows how many times to divide by the base.
Action Items / Next Steps
- Practice multiplying and dividing exponents with the same base.
- Remember to treat numbers and variables separately in combined terms.