Overview
This lecture explains the main laws of exponents, how to handle negative and zero exponents, rules for exponents with parentheses, and methods for simplifying roots and radicals. It also provides strategies for working through more complex expressions and common mistakes to avoid.
Laws of Exponents
- Any number or variable raised to the zero power equals one: ( a^0 = 1 ).
- A negative exponent means take the reciprocal: ( a^{-n} = \frac{1}{a^n} ).
- Example: Think of a negative exponent as "pushing" the base under a fraction bar, making the exponent positive.
- The root as an exponent: ( a^{1/n} = \sqrt[n]{a} ). The denominator of the exponent is the root.
- When multiplying like bases, add the exponents: ( x^m \times x^n = x^{m+n} ).
- When raising a power to another power, multiply the exponents: ( (x^m)^n = x^{mn} ).
- When multiplying different bases with the same exponent: ( (ab)^n = a^n b^n ).
- When dividing like bases, subtract the exponents: ( \frac{x^m}{x^n} = x^{m-n} ).
Working with Parentheses and Exponents
- If an exponent is outside parentheses and the inside is multiplication, multiply the exponents: ( (x^m)^n = x^{mn} ).
- If the inside of the parentheses is addition or subtraction, you must use distribution (not covered in this lecture).
- Negative exponents move the base to the denominator: ( x^{-n} = \frac{1}{x^n} ).
- A negative sign in front of an expression (not as an exponent) stays with the answer throughout the problem.
Simplifying Expressions
- Convert negative exponents to positive by moving the base to the denominator or numerator as needed.
- Combine like terms using exponent rules before applying exponents outside parentheses.
- To eliminate a negative exponent on a fraction, flip the entire fraction: ( \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n ).
- Simplify inside parentheses first, following the order of operations, before applying exponents outside.
Roots and Radicals
- The square root of ( a ) is written as ( a^{1/2} ); the denominator of the exponent indicates the root.
- The cube root of ( a ) is ( a^{1/3} ).
- Only perfect squares (like 4, 9, 16) or perfect cubes (like 8, 27) can be simplified directly under the root.
- You can add or subtract radicals only if the numbers under the root are the same: ( \sqrt{a} + \sqrt{a} = 2\sqrt{a} ).
- Always simplify inside the radical before combining terms.
Examples
- ( x^2 \times x^3 = x^{2+3} = x^5 )
- ( (x^2)^3 = x^{2 \times 3} = x^6 )
- ( x^{-1} = \frac{1}{x} )
- ( \sqrt{64} - \sqrt{81} = 8 - 9 = -1 )
- ( \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} )
- ( \sqrt{36} = 6 )
- ( \sqrt[3]{27} = 3 )
- ( \sqrt[3]{81} = \sqrt[3]{27 \times 3} = 3\sqrt[3]{3} )
- ( \sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6} )
- ( \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} )
- Note: You can only add or subtract radicals if the numbers under the root are the same.
Key Terms & Definitions
- Exponent — The power to which a number or variable is raised (e.g., in ( x^3 ), 3 is the exponent).
- Radical — The root symbol (√), used to denote roots such as square roots or cube roots.
- Base — The number or variable being raised to a power (e.g., in ( x^3 ), ( x ) is the base).
- Perfect Square — A number that is the square of an integer (e.g., 4, 9, 16).
- Perfect Cube — A number that is the cube of an integer (e.g., 8, 27, 64).
Action Items / Next Steps
- Practice simplifying exponent and radical expressions using these laws and rules.
- Review homework problems involving exponents and roots, focusing on converting negative exponents and simplifying radicals.
- Prepare questions for the next class if any concepts remain unclear, especially regarding negative exponents and combining radicals.