Chapter R Section 8: 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

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 R.8 nth Roots and Rational Expressions: nth Roots

where a, b are any real number if n is odd 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: R.8 nth Roots and Rational Expressions: nth Roots

Simplifying Radicals: A radical is in simplest form when: 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. R.8 nth Roots and Rational Expressions: Simplify Radical Expressions

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 R.8 nth Roots and Rational Expressions: Simplify Radical Expressions

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. R.8 nth Roots and Rational Expressions: Add, Subtract, Multiply and Divide Radicals

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

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

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. R.8 nth Roots and Rational Expressions: Rationalize Denominators

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. R.8 nth Roots and Rational Expressions: Rationalize Denominators

Evaluating Rational Exponents: Examples: Simplify each expression. R.8 nth Roots and Rational Expressions: Simplify Expressions with Rational Exponents

Simplifying Expressions Containing Rational Exponents: Recall the following from Review Section 2: Laws of Exponents: For any integers m, n (assuming no divisions by 0) and new! new! R.8 nth Roots and Rational Expressions: Simplify Expressions with Rational Exponents

Examples: Simplify each expression Examples: Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. R.8 nth Roots and Rational Expressions: Simplify Expressions with Rational Exponents

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. R.8 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational Exponents

Examples: Factor each expression Examples: Factor each expression. Express your answer so that only positive exponents occur. R.8 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational Exponents

Examples: Factor each expression Examples: Factor each expression. Express your answer so that only positive exponents occur. R.8 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational Exponents

Examples: Factor each expression Examples: Factor each expression. Express your answer so that only positive exponents occur. R.8 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational Exponents

where is the initial velocity in ft/sec of the object. 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? R.8 nth Roots and Rational Expressions: Applications

Independent Practice You learn math by doing math. The best way to learn math is to practice, practice, practice. The assigned homework examples provide you with an opportunity to practice. Be sure to complete every assigned problem (or more if you need additional practice). Check your answers to the odd-numbered problems in the back of the text to see whether you have correctly solved each problem; rework all problems that are incorrect. Read pp. 72-76 Homework: pp. 77-79 #7-51 odds, 55-73 odds, 89-93 odds, 107 Don’t leave your grades to chance…do your homework! R.8 nth Roots and Rational Expressions

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.” R.8 nth Roots and Rational Expressions