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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 10.3 Multiply Square Roots There are two ways that the product can be found. Let’s take a look at both ways. First method is to find the square roots of each and then multiply the results. Second method is to multiply the radicands and then find the square root. Notice that both will obtain the same results. This leads us to the product rule for Multiplying expressions containing square roots.

5. Example 1 Multiply. Solution Be Careful We can multiply radicals this way only if the indices are the same. We will see later how to multiply radicals with different indices such as

6. Simplify the Square Root of a Whole Number How do we know when a square root is simplified? To simplify expressions containing square roots, we reverse the process of multiplying. That is we use the product rule that says where a and b is a perfect square.

7. Example 2 Simplify completely. Solution a) The radical is not in simplest form since 45 contains a factor (other than 1) That is a perfect square. Think of two numbers that multiply to 45 so that at least One of the numbers is a perfect square: Notice, however that,but neither 3 or 15 is a perfect square. So we need to use 9 and 5. 9 is a perfect square. Product Rule. Examples b through d continued on next page…

8. Example 2 Simplify completely. Solution b) The radical is not in simplest form since 75 contains a factor (other than 1) Think of two numbers that multiply to 75 so that at least one of the numbers is a perfect square: Notice, 25 is a perfect square and 3 is not a perfect square. 25 is a perfect square. Product Rule. Examples c continued on next page…

9. Example 2 Simplify completely. Solution c) The radical is not in simplest form since 500 contains a factor (other than 1). Think of two numbers that multiply to 500 so that at least one of the numbers is perfect square: Notice, 100 is a perfect square and 5 is not a perfect square. 25 is a perfect square. Product Rule. Examples c continued on next page…

10. d) Does 48 have a factor that is a perfect square? 16 is a perfect square. Product Rule. However, notice that. Notice that is not in simplest form. We must continue to simplify. Example 2-Continued Simplify completely. Solution

11. Use the Quotient Rule for Square Roots Example 3 Simplify completely. Solution Quotient Rule Since 9 and 49 are perfect squares, find the square root of each separately.

12. Example 4 Simplify completely. Solution Simplify a) Neither 300 nor 3 are perfect squares, so you want to simplify to get 100, which is a perfect square. b) Neither 120 nor 10 are perfect squares. There are two methods that you can use to simplify One is to apply the quotient rule to obtain a fraction under one radical and then simplify and the second is to apply the product rule to rewrite each radical and then simplify the fraction. Quotient rule Square root of 4 is 2. Product rule Divide out common Factors. Method 2 Method 1

13. Example 4-Continued Simplify completely. Solution c) The fraction does not reduce. However, 36 is a perfect square. Begin by applying the quotient rule. Quotient rule

14. Multiplying is the same as dividing 6 by 2. We can generalize this result with The following statement. Simplify Square Root Expressions Containing Variables with Even Exponents A square root is not simplified if it contains any factors that are perfect squares. This means that a square rot containing variables is simplified if the power on each variable is less than 2. For example, is not in simplified form. If r represents a nonnegative real number, then we can use rational exponents to simplify. We can combine this property with the product and quotient rules to simplify radical expressions.

15. Example 5 Simplify completely. Solution Product rule 4 is a perfect square. Simplify. Rewrite using the commutative property. Begin by using the quotient rule.

16. Simplify Square Root Expressions Containing Variables with Odd Exponents How do we simplify an expression containing a square root if the power under the Square root is odd? We can use this product rule for radicals and fractional exponents to help us understand how to simplify such expressions. Example 6 Simplify completely. Solution To simplify the radicals, write the variable as the product of two factors so that the exponent of one of the factors is the largest numbers less than 9 that is divisible by 2 (the index of the radical). 8 is the largest number less than 9 that is divisible by 2. Product Rule Use a fractional exponent to simplify. 12 is the largest number less than 9 that is divisible by 2. Product Rule Use a fractional exponent to simplify.

17. If you notice in the previous examples, we always divided by 2. Let’s look at the previous examples to see if we can used division to help us simplify radicals. Quotient Remainder Index of radical Quotient Remainder Index of radical

18. Example 7 Simplify completely. Solution Product Rule Use commutative property to rewrite expression Use the product rule to write the expression with one radical.

19. Example 8 Simplify completely. Solution Use the Product Rule for each Radical. Use commutative property to rewrite expression Use the product rule to write the expression with one radical. Quotient Rule Product Rule Use commutative property to rewrite expression Use the product rule to write the expression with one radical.

20. Simplify More Square Root Expressions Containing Variables Example 9 Simplify completely. Solution Product Rule Multiply the radicands together to obtain one radical. Product Rule Evaluate Commutative property

21. Example 9-Continued Simplify completely. Solution Use Quotient Rule. Product Rule. Commutative property