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Algebra 2 Notes Name: ________________ Section 6.6 – Radical Expressions & Radical Exponents You are probably familiar with finding the square and square root of a number. These two operations are inverses of each other. Similarly, there are roots that correspond to larger powers. 5 and 5 are _________________ roots of 25 because 52 25 and 5 25 . 2 2 is the _________________ root of 8 because 2 3 8 . 4 2 and 2 are _________________ roots of 16 because 2 4 16 and 2 16 . a is the _________________ roots of b if a n b . The n th root of a real number a can be written as the radical expression _________________, where n is the _________________ of the radical and a is the _________________. When a number has more than one real root, the radical sign indicates only the _________________, or positive, root. Case Odd index Even index; positive radicand Even index; negative radicand Radicand of 0 Numbers and Types of Real Roots Roots Example 1 real root The real 3rd root of 8 is 2. 2 real roots The real 4th roots of 16 are 0 real roots -16 has no real 4th root. 1 root of 0 The 3rd root of 0 is 0. Example 1: Find all real roots. a. fourth roots of 81 b. cube roots of -125 The properties of square roots also apply to Properties of For a 0 and a. 3 27x6 n th roots. n th Roots b 0, Words Product Property of Roots The n th root of a product is equal to the product of the n n roots. Quotient Property of Roots The n th root of a quotient is equal to the quotient of the n th root. Example 2: c. square root of –25 Numbers 3 Algebra 16 3 8 3 2 2 3 2 n ab n a n b 25 25 5 16 16 4 n a na b nb Simplify each expression. Assume that all variables are positive. b. 4 3 x8 c. 3 x3 7 d. 3 x7 3 x2 2. A ____________________ exponent is an exponent that can be expressed as _______________, where m and n are integers and n 0 . ____________________ expressions can be written by using rational exponents. Rational Exponents For any natural number n and integer Words m, Numbers 1 4 1 The exponent indicates the n th root. n m The exponent indicated the n th root n raised to the m th power. Example 3: a. 125 a. 7 1 n 16 16 2 2 83 a na 4 3 2 8 m 22 4 an a n m n am Write the expression in radical form and then simplify. 2 3 Example 4: 4 Algebra 1 b. 1 5 c. 64 3 d. 42 32 5 Rewrite each expression using rational exponents. Then simplify when possible. b. 3 3 116 c. 4 252 Rational exponents have the same properties as integer exponents. Look back in Section 1.5 in your book if you need to remember… Example 5: a. 3 5 25 25 Simplify each expression. 2 5 3 1 b. 83 2 83 c. 1 36 8 36 8