2 Chapter R: Reference: Basic Algebraic Concepts R.1 Review of Exponents and PolynomialsR.2 Review of FactoringR.3 Review of Rational ExpressionsR.4 Review of Negative and Rational ExponentsR.5 Review of Radicals
3 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
4 R.5 Review of RadicalsIn the expressionis called a radical sign,a is called the radicand,n is called the index.
5 R.5 Evaluating Roots Example Evaluate each root. (a) (b) (c) Solution (b) is not a real number.(c)
6 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
7 R.5 Converting from Rational Exponents to Radicals Example Write in radical form and simplify.(a) (b) (c)Solution(a)(b)(c)
8 R.5 Converting from Radicals to Rational Exponents Example Write in exponential form.(a) (b) (c)Solution(a) (b)(c)
9 R.5 Review of Radicals Evaluating If n is an even positive integer, thenIf n is an odd positive integer, then
10 R.5 Using Absolute Value to Simplify Roots Example Simplify each expression.(a) (b) (c)Solution(a)(b)(c)
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,
12 R.5 Using the Rules for Radicals to Simplify Radical Expressions Example Simplify each expression.(a) (b) (c)Solution(a)(b)(c)
13 R.5 Simplifying Radicals Simplified RadicalsAn 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).
14 R.5 Simplifying Radicals Example Simplify each radical.(a) (b)Solution(a)(b)
15 R.5 Simplifying Radicals by Writing Them with Rational Exponents Example Simplify each radical.(a) (b)Solution(a)(b)
16 R.5 Adding and Subtracting Like Radicals Example Add or subtract, as indicated. Assume allvariables represent positive real numbers.(a) (b)Solution(a)
17 R.5 Adding and Subtracting Like Radicals Solution (b)
18 R.5 Multiplying Radical Expressions Example Find each product.(a) (b)Solution (a) Using FOIL,
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.
21 R.5 Rationalizing Denominators Example Rationalize each denominator.(a) (b)Solution(a)(b)
22 R.5 Rationalizing a Binomial Denominator Example Rationalize the denominator ofSolution