2. Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example 1: 4 Example 2: Example 3:

3. Product Rule Simplify radicals Multiply Coefficients Multiply radicands –“Roots” must be the same Simplify, if needed

4. Examples: Product Rule

5. Quotient Rule Fractions made up of radicals can be simplified just like fractions 2

7. Multiply the radicals. 1. Simplify.

8. 6. Simplify. Multiply the coefficients and the radicals.

9. 7. Simplify. Divide the radicals.

10. 8. Simplify.

11. Rationalizing Radicals To simplify a fraction with a radical in the denominator, multiply the numerator and denominator by the radical. Example 1: Estimation is easier with rational denominators. This process is called rationalizing the denominator. Example 2: Since the square root of a quotient is a quotient of square roots, the square root of a fraction must be rationalized to be in simplest form.

12. Answer: 9. Simplify.

13. Radicals representing square roots of different numbers can not be gathered like this. Adding and Subtracting Radicals Radicals that represent the square root of the same number can be treated as a common factor. Examples: But simplifying sometimes results in multiples of the same radical, which can be. Examples: Like terms can be gathered. Unlike terms can not.

14. Combining Like Terms Radicals & Like Terms –Same variables –Variables have the same exponents –IDENTICAL RADICALS Examples

15. Simplify radicals if possible Combine coefficients Radicals ARE simplified

16. 1. Simplify. Just like when adding variables, you can only combine LIKE radicals. 5 √5

17. Answer: 4 √7 +3 √3 2. Simplify.

18. Simplify each radical. 4√93 - 2√16 3 + 2√45 4 3√3 - 2 4√3+2 2√5 12√3 - 8√3 + 4√5 Combine like radicals 4√3 + 4√5 3. Simplify.

19. More Radical Fun SIMPLIFY MULTIPLY Must have Common Denominators

20. Distributive Property with Radicals

21. Multiplying Binomials With Radicals Multiplying binomials that contain radicals sometimes results in products of radicals that can be simplified. Examples: 9 - 54 1. 2.3.

22. Conjugates Binomials of the form and that are identical except for the sign separating the terms are called conjugates. Multiplying conjugates like these together results in a rational number: Conjugates are therefore used to rationalize certain fractions. Example: a 2 b - c 2 d

25. Practice Multiply: Divide: Add: Subtract: Multiply: Rationalize: 15 0