Overview
This lecture covers how to evaluate and simplify radical expressions, use rational exponents, perform operations with radicals, and rationalize denominators.
Evaluating Square Roots
- The square root of a number is the nonnegative value that, when squared, gives the original number.
- The principal square root is always nonnegative and denoted with the radical symbol (√).
- Example: √100 = 10, √16 = 4, √(25+144) = √169 = 13.
Product and Quotient Rules for Square Roots
- Product Rule: √(ab) = √a × √b, for nonnegative a and b.
- To simplify √(300), factor perfect squares and apply the product rule.
- Quotient Rule: √(a/b) = √a / √b, where b ≠0.
- Simplify numerators and denominators separately when using the quotient rule.
Operations with Square Roots
- Addition/Subtraction: Combine radicals only if they have the same radicand and index.
- Simplify first if possible, then add or subtract terms with like radicals.
- Example: 2√2 + 3√2 = 5√2.
Rationalizing Denominators
- To remove a radical from a denominator, multiply numerator and denominator by a value that will eliminate the radical.
- For denominators with two terms (one rational, one irrational), multiply by the conjugate.
Using Rational Roots and nth Roots
- The nth root of a is a number that, raised to the nth power, gives a.
- The index in a radical denotes which root; if no index, it's a square root.
- Example: ∛8 = 2, since 2³ = 8.
Rational Exponents
- A rational exponent translates a radical: a^(1/n) = √[n]{a}.
- For a^(m/n), raise a to the mth power then take the nth root: a^(m/n) = (√[n]{a})^m.
- All rules of exponents apply to rational exponents.
Key Terms & Definitions
- Radical — The symbol (√) used to denote roots.
- Radicand — The number under the radical symbol.
- Principal Square Root — The nonnegative square root of a number.
- Conjugate — A binomial formed by changing the sign between two terms, used to rationalize denominators.
- nth Root — A value that, when raised to the n power, gives the original number.
- Rational Exponent — An exponent in fractional form, expressing roots and powers.
Action Items / Next Steps
- Complete the "Try It" exercises and section exercises for additional practice.
- Review all key terms and practice simplifying radical and rational exponent expressions.
- Prepare for real-world application problems involving the Pythagorean Theorem and roots.