# Chapter R: Reference: Basic Algebraic Concepts

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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

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

R.5 Review of Radicals In the expression is called a radical sign, a is called the radicand, n is called the index.

R.5 Evaluating Roots Example Evaluate each root. (a) (b) (c) Solution
(b) is not a real number. (c)

If a is a real number, m is an integer, n is a positive integer, and is a real number, then

R.5 Converting from Rational Exponents to Radicals
Example Write in radical form and simplify. (a) (b) (c) Solution (a) (b) (c)

R.5 Converting from Radicals to Rational Exponents
Example Write in exponential form. (a) (b) (c) Solution (a) (b) (c)

If n is an even positive integer, then If n is an odd positive integer, then

R.5 Using Absolute Value to Simplify Roots
Example Simplify each expression. (a) (b) (c) Solution (a) (b) (c)

For all real numbers a and b, and positive integers m and n for which the indicated roots are real numbers,

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

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).

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

R.5 Simplifying Radicals by Writing Them with Rational Exponents
Example Simplify each radical. (a) (b) Solution (a) (b)

Example Add or subtract, as indicated. Assume all variables represent positive real numbers. (a) (b) Solution (a)

Solution (b)

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

Solution (b)

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.

R.5 Rationalizing Denominators
Example Rationalize each denominator. (a) (b) Solution (a) (b)

R.5 Rationalizing a Binomial Denominator
Example Rationalize the denominator of Solution