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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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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

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**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

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**Evaluating Rational Exponents:**

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

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**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

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**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

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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

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**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

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**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

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**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

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**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

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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 Homework: pp #7-51 odds, odds, 89-93 odds, 107 Don’t leave your grades to chance…do your homework! R.8 nth Roots and Rational Expressions

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**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

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To divide radicals: divide the coefficients divide the radicands if possible rationalize the denominator so that no radical remains in the denominator.

To divide radicals: divide the coefficients divide the radicands if possible rationalize the denominator so that no radical remains in the denominator.

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