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2. Chapter 10 Radicals and Rational Exponents

3. 10.1 Finding Roots 10.2 Rational Exponents 10.3 Simplifying Expressions Containing Square Roots 10.4Simplifying Expressions Containing Higher Roots 10.5Adding, Subtracting, and Multiplying Radicals 10.6Dividing Radicals Putting it All Together 10.7Solving Radical Equations 10.8Complex Numbers 10 Radicals and Rational Exponents

4. Simplifying Expressions Containing Square Roots 10.4 Multiply Higher Roots This rule enables us to multiply and simplify radicals with any index in a way that is similar to multiplying and simplifying square roots. Be Careful Remember that we can apply the product rule in this way only if the indices Of the radicals are the same. Later in this section, we will discuss how to Multiply radicals with different indices.

5. Example 1 Multiply. Solution

6. Simplify Higher Roots of Integers We saw in the previous section that a simplified square root cannot contain any perfect squares. Next we will list the conditions that determine when a radical with any index is in simplest form. Note Condition 1) implies that the radical cannot contain variables with exponents Greater than or equal to n, the index of the radical. To simplify radicals with any index, use the product rule,where a or b is an nth power.

7. Example 2 Simplify completely. Solution a) We must simplify the cube root of 81, think of two numbers that multiply to 81 so that at least one of the numbers is a perfect cube: 27 is a perfect cube. Product Rule. b) We must simplify the fourth root of 144, think of two numbers that multiply to 144 so that at least one of the numbers is a perfect fourth power:

8. Use the Quotient Rule for Higher Roots We will apply the quotient rule when working with nth roots the same way we apply It when working with square roots. Example 3 Simplify completely. Solution

9. Example 4 Simplify completely. Solution Let’s begin by applying the quotient rule to obtain a fraction under one radical, then simplifying the fraction. Quotient rule. Simplify. 8 is a perfect cube. Product Rule.

10. Simplify Radicals Containing Variables This is the principle we use to simplify radicals with indices greater than 2. Example 5 Simplify completely. Solution Product rule. Write with rational exponents. Simplify exponents.

11. Example 6 Simplify completely. Solution Quotient rule. Use Rational Exponents

12. Example 7 Simplify completely. Solution Notice that 31 does not divide evenly by the index. Therefore, to simplify write as The product of two factors so that the exponent of one of the factors is the largest Number less than 31 that is divisible by 4 (the index). 28 is the largest number less than 31 that is divisible by 4. Product rule. Use fractional exponent to simplify.

13. Example 8 Simplify completely. Solution Product rule. Use fractional exponent to simplify. Commutative Property to rewrite expression.

14. Multiply and Divide Radicals with Different Indices The product and quotient rules for radicals only apply when the radicals have the same indices. To multiply or divide radicals with different indices, we first change the radical expressions to rational exponent form. Example 9 Simplify completely. Solution Change radical to fractional expressions. Get a common denominator to add exponents. Add exponents. Rewrite in radical form and simplify.