Overview
This lecture covers the properties and operations of radicals and rational exponents, including simplifying, evaluating, and manipulating radical expressions, and converting between radicals and rational exponents.
Evaluating Square Roots
- The square root of a number is the value that, when squared, equals the original number.
- The principal square root is the nonnegative square root, denoted by the radical symbol √.
- Only the principal root is returned by calculators.
Simplifying Square Roots: Product & Quotient Rules
- The product rule: √(ab) = √a × √b for any nonnegative a, b.
- To simplify, factor perfect squares and rewrite the expression using the product rule.
- The quotient rule: √(a/b) = √a / √b for b ≠0.
- Simplify numerator and denominator separately before dividing.
Adding and Subtracting Square Roots
- Add or subtract radicals only if they have the same radicand and index.
- Simplify radicals first to find like terms, then combine.
Rationalizing Denominators
- Expressions are simplified by removing radicals from denominators.
- For a single radical denominator, multiply numerator and denominator by the radical.
- For a denominator with two terms (one involving a radical), multiply numerator and denominator by the conjugate.
Using Rational Roots and n-th Roots
- n-th roots (e.g., cube root, fourth root) are inverse operations of raising to the nth power.
- The n-th root of a, written as √[n]{a}, is the number that raised to the n-th power equals a.
- The index of the radical is n.
Rational Exponents
- Radical expressions can be rewritten with rational exponents: a^(1/n) = √[n]{a}.
- For exponents m/n, a^(m/n) = (√[n]{a})^m = √[n]{a^m}.
- Rational exponents follow all the usual exponent rules.
Key Terms & Definitions
- Radical — The symbol √ used to denote roots.
- Radicand — The value under the radical sign.
- Principal Square Root — The nonnegative root of a number.
- Product Rule (Radicals) — √(ab) = √a × √b.
- Quotient Rule (Radicals) — √(a/b) = √a / √b.
- Conjugate — For a+b√c, the conjugate is a−b√c.
- n-th Root — The value that raised to the n-th power equals the original number.
- Rational Exponent — An exponent expressed as a fraction, representing both a root and a power.
Action Items / Next Steps
- Complete "Try It" exercises and section exercises from the textbook.
- Review the definitions and rules for radicals and rational exponents.
- Practice converting between radical and rational exponent forms.
- Prepare for exercises involving adding, subtracting, and rationalizing radical expressions.