Overview
This lecture summarizes the key laws of exponents and radicals, including how to simplify and manipulate powers, handle negative and zero exponents, and use rational exponents.
Laws of Exponents
- Multiplication law: When multiplying like bases, add the exponents (e.g., ( x^2 \times x^3 = x^{2+3} = x^5 )).
- Division law: When dividing like bases, subtract the exponents (e.g., ( x^3 / x^2 = x^{3-2} = x^1 )).
- Power law: When raising a power to another power, multiply the exponents (e.g., ( (x^3)^2 = x^{3 \times 2} = x^6 )).
- Distributive law: Exponents distribute over multiplication and division, not over addition (e.g., ( (xy)^a = x^a y^a ), but ( (x+y)^a \neq x^a + y^a )).
- Zero exponent: Any nonzero base to the 0 power equals 1 (e.g., ( x^0 = 1 )).
- Negative exponent: A negative exponent indicates the reciprocal (e.g., ( x^{-a} = 1/x^a )).
- Fractional exponents: ( x^{1/m} ) is the m-th root of x; ( x^{m/n} ) is the n-th root of ( x^m ).
Key Properties & Examples
- Do not multiply bases when multiplying powers; only the exponents are added if bases are identical.
- When exponents are negative, move the base to the denominator or numerator to make the exponent positive.
- Coefficients are not affected by exponent rules unless they are included in the base.
- Simplifying fractional and negative exponents involves applying the above laws step by step.
- When simplifying, always aim for positive exponents, unless specified otherwise.
Mixed Examples
- ( b^5 \times b / b^2 = b^{5+1-2} = b^4 )
- ( (y^2)^3 / y^6 = y^{6-6} = y^0 = 1 )
- ( (2a^4)^3 \times a^3^2 = 8a^{12+6} = 8a^{18} )
- ( (2/3x^2y)^{-3} = -27y^3/8x^6 )
- ( 25^{1/2} = 5 ); ( 36^{3/2} = 216 ); ( (4x^3)^{1/2} = 2x\sqrt{x} )
Key Terms & Definitions
- Exponent — The number indicating how many times a base is multiplied by itself.
- Base — The number or expression being raised to a power.
- Power — The result of raising a base to an exponent.
- Monomial — An algebraic expression of one term.
- Binomial — An algebraic expression of two terms.
- Reciprocal — The inverse of a number, ( 1/x ).
- Radical — The root expression, e.g., ( \sqrt{x} ).
Action Items / Next Steps
- Practice simplifying expressions using all exponent laws.
- Complete assigned homework on exponent and radical laws.
- Review mistakes related to negative and zero exponents.
- Prepare for next lesson on further radical expressions.