Overview
This lecture covers the rules and properties of zero exponents, negative exponents, and rational (fractional) exponents, with step-by-step examples for each concept.
Zero Exponents
- Any number, variable, or expression raised to the power of zero equals one (e.g., ( x^0 = 1 ), ( 3^0 = 1 )).
- When multiplying ( 3^0 ) by ( y ), result is simply ( y ) as ( 1 \times y = y ).
- Any variable without an exponent is understood to have an exponent of one.
- For sums, ( 5^0 + 3 = 1 + 3 = 4 ).
- For expressions, ( (x+25)^0 = 1 ); if combined with other terms, follow order of operations.
Negative Exponents
- A negative exponent means take the reciprocal and change the exponent to positive (( a^{-n} = 1/a^n )).
- If negative exponent is in the denominator (( 1/a^{-n} )), move to numerator as positive (( a^n )).
- Example: ( 2^{-3} = 1/2^3 = 1/8 ).
- Example: ( 1/3^{-2} = 3^2 = 9 ).
- Negative signs on numbers themselves (not exponents) do not affect exponent rules.
- Multiply or combine like bases using exponent addition, even after moving terms.
Rational Exponents (Fractional Exponents)
- Rational exponents are fractions: ( a^{p/q} ) means the q-th root of a raised to the p (( (a^p)^{1/q} ) or ( \sqrt[q]{a^p} )).
- When an expression with a rational exponent is raised to another power, multiply the exponents.
- Example: ( 3^{1/2} ) raised to 6: ( (3^{1/2})^6 = 3^{3} = 27 ).
- Combine exponents before simplifying: ( 5^{2/3} ) raised to 3: ( 5^{2} = 25 ).
- Distribute exponents across multiplication: ( (2^6 x^8 y^4)^{1/2} = 2^3 x^4 y^2 = 8x^4y^2 ).
Key Terms & Definitions
- Zero Exponent — Any base (except zero) raised to the zero power equals one.
- Negative Exponent — Indicates reciprocal; ( a^{-n} = 1/a^n ).
- Rational Exponent — An exponent in fraction form; indicates roots and powers.
- Reciprocal — Flipping a number over (e.g., reciprocal of ( a ) is ( 1/a )).
Action Items / Next Steps
- Complete the assigned practice problems (examples marked in lecture) and submit answers as instructed.
- Review the laws of exponents, focusing on product, power, and quotient rules.
- Prepare questions on zero, negative, and rational exponents for next class discussion.