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Radicals and Rational Exponents Chapter 10

10 Radicals and Rational Exponents 10.1 Finding Roots 10.3 Simplifying Expressions Containing Square Roots 10.4 Simplifying Expressions Containing Higher Roots 10.5 Adding, Subtracting, and Multiplying Radicals 10.6 Dividing Radicals Putting it All Together 10.7 Solving Radical Equations 10.8 Complex Numbers

10.6 Dividing Radicals To eliminate a radical from its denominator we use a process called rationalizing the denominator. We will be looking at two types of rationalization problems. 1) Rationalize a denominator containing one term. 2) Rationalize a denominator containing two terms. We will use the idea that multiplying a numerator and denominator of a fraction by The same quantity results in an equivalent fraction.

Rationalize a Denominator: One Square Root The goal of rationalizing is to eliminate the radical from the denominator. Recall that, We will use this property to rationalize denominator. Example 1 Rationalize the denominator of each expression. Solution Rationalize denominator

Rationalize the denominator of each expression. Note Incorrect! Example 2 Rationalize the denominator of each expression. Solution To rationalize the denominator of begin by simplifying . Simplify square root of 20. Rationalize denominator

Rationalize the denominator of each expression. Example 3 Rationalize the denominator of each expression. Solution Rationalize denominator

To Simplify we must rationalize the denominator. Example 4 Simplify completely. Solution To Simplify we must rationalize the denominator. Rationalize denominator Rationalize denominator

Example 5 Simplify Completely. Solution Simplify the radicand using the quotient rule for exponents. After simplifying now we can rationalize the denominator. Simplify Rationalize denominator Multiply

Rationalize a Denominator: One Higher Root Many student assume that to rationalize denominators we simply multiply the denominator and numerator of the expression by the denominator as in the previous section. Let’s practice eliminating radicals before we move on to rationalizing. Example 6 Fill in the blank. Solution

Example 7 Fill in the blank. Solution

Rationalize the denominator. Example 8 Rationalize the denominator. Solution a) First identify what we want the denominator to be after multiplying. Rationalize denominator Multiply Simplify Rationalize denominator Multiply Simplify

Rationalize the denominator. Example 9 Rationalize the denominator. Solution a) First identify what we want the denominator to be after multiplying. Rationalize denominator Multiply Simplify

Rationalize a Denominator: Containing Two Terms

Each method, i and ii, both give the same result. Example 10 Solution i) Use FOIL to multiply. F O I L Each method, i and ii, both give the same result. ii) Use

Example 11 Rationalize the denominator and simplify completely. Solution Multiply by the conjugate. Use FOIL or Simplify. Subtract.

Example 12 Rationalize the denominator and simplify completely. Solution Multiply by the conjugate. Use FOIL or in denominator. Use FOIL in the numerator. Simplify.

Rationalize a Numerator In higher-level math courses, sometimes it is necessary to rationalize the numerator of a radical expression so that the numerator does not contain a radical. Example 13 Rationalize the numerator and simplify completely. Solution Rationalize Numerator. Multiply Simplify.

Divide Out Common Factors from the Numerator and Denominator Sometimes it is necessary to simplify a radical expression by dividing out common factors from the numerator and denominator. This is a skill we will need in chapter 11 to solve Quadratic equations, so we look at an example here. Example 14 Simplify completely. Solution It is tempting to do one of the following: Each is incorrect because is a term in a sum and 18 is a term in a sum. Be Careful Incorrect! Incorrect! The correct way to simplify is to begin by factoring out a 9 in the numerator and then divide the numerator and denominator by any common factors. We will see this on the next slide.

Factor out 9 from the numerator. Divide by 9. Simplify.