In this section, we will… Evaluate nth Roots Simplify Radical Expressions Add, Subtract, Multiply and Divide Radi

1. 7.1 nth Roots and Rational Exponents • In this section, we will… • Evaluate nth Roots • Simplify Radical Expressions • Add, Subtract, Multiply and • Divide Radical Expressions • Rationalize Denominators • Simplify Expressions with Rational Exponents • Factor Expressions with Radicals or Rational Exponents

2. Recall from Review Section 2… If a is a non-negative real number, any number b, such that is the square root of a and is denoted If a is a non-negative real number, any non-negative number b, such that is the primary square root of a and is denoted Examples: Evaluate the following by taking the square root. The principal root of a positive number is positive principal root: Negative numbers do not have real # square roots 7.1 nth Roots and Rational Expressions: nth Roots

3. The principal nth root of a real number a, n > 2 an integer, symbolized by is defined as follows: • where and if n is even • where a, b are any real number if n is odd Examples: Simplify each expression. index radicand radical principal root: 7.1 nth Roots and Rational Expressions: nth Roots

4. Properties of Radicals: Let and denote positive integers and let a and b represent real numbers. Assuming that all radicals are defined: Simplifying Radicals: A radical is in simplest form when: • No radicals appear in the denominator of a fraction • The radicand cannot have any factors that are perfect roots (given the index) Examples: Simplify each expression. 7.1 nth Roots and Rational Expressions: nth Roots

5. Simplifying Radical Expressions Containing Variables: Examples: Simplify each expression. Assume that all variables are positive. When we divide the exponent by the index, the remainder remains under the radical 7.1 nth Roots and Rational Expressions: nth Roots

6. Adding and Subtracting Radical Expressions: • simplify each radical expression • combine all like-radicals (combine the coefficients and keep the common radical) Examples: Simplify each expression. Assume that all variables are positive. 7.1 nth Roots and Rational Expressions: nth Roots

7. Multiplying and Dividing Radical Expressions: Examples: Simplify each expression. Assume that all variables are positive. 7.1 nth Roots and Rational Expressions: Add, Subtract, Multiply and Divide Radicals

8. Examples: Simplify each expression. Assume that all variables are positive. 7.1 nth Roots and Rational Expressions: Add, Subtract, Multiply and Divide Radicals

9. Rationalizing Denominators: Recall that simplifying a radical expression means that no radicals appear in the denominator of a fraction. Examples: Simplify each expression. Assume that all variables are positive. 7.1 nth Roots and Rational Expressions: Rationalize Denominators

10. Rationalizing Binomial Denominators: example: Examples: Simplify each expression. Assume that all variables are positive. The conjugate of the binomial a + b is a – b and the conjugate of a – b is a + b. 7.1 nth Roots and Rational Expressions: Rationalize Denominators

11. Evaluating Rational Exponents: Examples: Simplify each expression. 7.1 nth Roots and Rational Expressions: Simplify Expressions with Rational Exponents

12. Simplifying Expressions Containing Rational Exponents: Recall the following from Review Section 2: and new! new! 7.1 nth Roots and Rational Expressions: Simplify Expressions with Rational Exponents

13. Examples: Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. 7.1 nth Roots and Rational Expressions: Simplify Expressions with Rational Exponents

14. Factoring Expressions with Radicals and/or Rational Exponents: Recall that, when factoring, we take out the GCF with the smallest exponent in the terms. Examples: Factor each expression. Express your answer so that only positive exponents occur. 7.1 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational Exponents

15. Examples: Factor each expression. Express your answer so that only positive exponents occur. 7.1 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational Exponents

16. Examples: Factor each expression. Express your answer so that only positive exponents occur. 7.1 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational Exponents

17. Examples: Factor each expression. Express your answer so that only positive exponents occur. 7.1 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational Exponents

18. Example: The final velocity, v, of an object in feet per second (ft/sec) after it slides down a frictionless inclined plane of height h feet is: where is the initial velocity in ft/sec of the object. What is the final velocity, v, of an object that slides down a frictionless inclined plane of height 2 feet with an initial velocity of 4 ft/sec? 7.1 nth Roots and Rational Expressions: Applications

19. Homework: pp. 404 Q 13 – 64 Read pp. 407 - 410 7.1 nth Roots and Rational Expressions

20. Review of Exam Policies and Procedures Page 7 of the Student Guide and Syllabus From Math for Artists… “These are the laws of exponents and radicals in bright, cheerful, easy to memorize colors.” 7.1 nth Roots and Rational Expressions