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**Chapter R: Reference: Basic Algebraic Concepts**

R.1 Review of Exponents and Polynomials R.2 Review of Factoring R.3 Review of Rational Expressions R.4 Review of Negative and Rational Exponents R.5 Review of Radicals

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**R.5 Review of Radicals Radical Notation for a1/n**

If a is a real number, n is a positive integer, and a1/n is a real number, then

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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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**R.5 Evaluating Roots Example Evaluate each root. (a) (b) (c) Solution**

(b) is not a real number. (c)

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**R.5 Review of Radicals Radical Notation for am/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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**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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**R.5 Converting from Radicals to Rational Exponents**

Example Write in exponential form. (a) (b) (c) Solution (a) (b) (c)

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**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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**R.5 Using Absolute Value to Simplify Roots**

Example Simplify each expression. (a) (b) (c) Solution (a) (b) (c)

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**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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**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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**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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**R.5 Simplifying Radicals**

Example Simplify each radical. (a) (b) Solution (a) (b)

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**R.5 Simplifying Radicals by Writing Them with Rational Exponents**

Example Simplify each radical. (a) (b) Solution (a) (b)

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**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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**R.5 Adding and Subtracting Like Radicals**

Solution (b)

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**R.5 Multiplying Radical Expressions**

Example Find each product. (a) (b) Solution (a) Using FOIL,

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**R.5 Multiplying Radical Expressions**

Solution (b)

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**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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**R.5 Rationalizing Denominators**

Example Rationalize each denominator. (a) (b) Solution (a) (b)

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**R.5 Rationalizing a Binomial Denominator**

Example Rationalize the denominator of Solution

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