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Copyright © 2007 Pearson Education, Inc. Slide R-2 Chapter R: Reference: Basic Algebraic Concepts R.1Review of Exponents and Polynomials R.2Review of Factoring R.3Review of Rational Expressions R.4Review of Negative and Rational Exponents R.5Review of Radicals

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Copyright © 2007 Pearson Education, Inc. Slide R-3 R.5 Review of Radicals Radical Notation for a 1/n If a is a real number, n is a positive integer, and a 1/n is a real number, then

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Copyright © 2007 Pearson Education, Inc. Slide R-4 R.5 Review of Radicals In the expression is called a radical sign, a is called the radicand, n is called the index.

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Copyright © 2007 Pearson Education, Inc. Slide R-5 R.5 Evaluating Roots Example Evaluate each root. (a) (b)(c) Solution (a) (b) is not a real number. (c)

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Copyright © 2007 Pearson Education, Inc. Slide R-6 R.5 Review of Radicals Radical Notation for a m/n If a is a real number, m is an integer, n is a positive integer, and is a real number, then

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Copyright © 2007 Pearson Education, Inc. Slide R-7 R.5 Converting from Rational Exponents to Radicals Example Write in radical form and simplify. (a) (b)(c) Solution (a) (b) (c)

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Copyright © 2007 Pearson Education, Inc. Slide R-8 R.5 Converting from Radicals to Rational Exponents Example Write in exponential form. (a) (b)(c) Solution (a)(b) (c)

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Copyright © 2007 Pearson Education, Inc. Slide R-9 R.5 Review of Radicals Evaluating If n is an even positive integer, then If n is an odd positive integer, then

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Copyright © 2007 Pearson Education, Inc. Slide R-10 R.5 Using Absolute Value to Simplify Roots Example Simplify each expression. (a) (b)(c) Solution (a) (b) (c)

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Copyright © 2007 Pearson Education, Inc. Slide R-11 R.5 Review of Radicals Rules for Radicals For all real numbers a and b, and positive integers m and n for which the indicated roots are real numbers,

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Copyright © 2007 Pearson Education, Inc. Slide R-12 R.5 Using the Rules for Radicals to Simplify Radical Expressions Example Simplify each expression. (a) (b)(c) Solution (a) (b) (c)

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Copyright © 2007 Pearson Education, Inc. Slide R-13 R.5 Simplifying Radicals Simplified Radicals An expression with radicals is simplified when the following conditions are satisfied. 1. The radicand has no factor raised to a power greater than or equal to the index. 2. The radicand has no fractions. 3. No denominator contains a radical. 4. Exponents in the radicand and the index of the radical have no common factor. 5. All indicated operations have been performed (if possible).

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Copyright © 2007 Pearson Education, Inc. Slide R-14 R.5 Simplifying Radicals Example Simplify each radical. (a) (b) Solution (a) (b)

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Copyright © 2007 Pearson Education, Inc. Slide R-15 R.5 Simplifying Radicals by Writing Them with Rational Exponents Example Simplify each radical. (a) (b) Solution (a) (b)

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Copyright © 2007 Pearson Education, Inc. Slide R-16 R.5 Adding and Subtracting Like Radicals Example Add or subtract, as indicated. Assume all variables represent positive real numbers. (a) (b) Solution (a)

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Copyright © 2007 Pearson Education, Inc. Slide R-17 R.5 Adding and Subtracting Like Radicals Solution (b)

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Copyright © 2007 Pearson Education, Inc. Slide R-18 R.5 Multiplying Radical Expressions Example Find each product. (a) (b) Solution (a) Using FOIL,

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Copyright © 2007 Pearson Education, Inc. Slide R-19 R.5 Multiplying Radical Expressions Solution (b)

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Copyright © 2007 Pearson Education, Inc. Slide R-20 R.5 Rationalizing Denominators The process of simplifying a radical expression so that no denominator contains a radical is called rationalizing the denominator. Rationalizing the denominator is accomplished by multiplying by a suitable form of 1.

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Copyright © 2007 Pearson Education, Inc. Slide R-21 R.5 Rationalizing Denominators Example Rationalize each denominator. (a) (b) Solution (a) (b)

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Copyright © 2007 Pearson Education, Inc. Slide R-22 R.5 Rationalizing a Binomial Denominator Example Rationalize the denominator of Solution

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