1. Simplifying Radicals: Part I  T T o simplify a radical in which the radicand contains a perfect square as a factor Example: √729 √9 ∙√81 3 ∙ 9 Perfect Square! 27

2. Vocabulary and Key Concepts Radical symbol Radicand Read “the square root of x.” NOTE: The index 2 is usually omitted when writing square roots. 2 Index

3. Table of Perfect Squares 1 2 = _____6 2 = _____11 2 = _____16 2 = _____ 2 2 = _____7 2 = _____12 2 = _____17 2 = _____ 3 2 = _____8 2 = _____13 2 = _____18 2 = _____ 4 2 = _____9 2 = _____14 2 = _____19 2 = _____ 5 2 = _____10 2 = _____15 2 = _____20 2 = _____ 1 36 121 256 4 49 144 289 9 64 169 324 16 81 196 361 25 100 225 400 Complete the table below: You may find the following table of perfect squares to be helpful when you are required to simplify square roots.

4. MENTAL MATH: Find two factors of 72, one of which is the greatest perfect square factor. Establish order, so that you don’t omit any! 1, 72 2,36 3, 24 4, 18 6, 12 8, 9 9,8 (once you have a repeated factor pair, you know that you have found ALL factors!) Simplifying Square Roots ALERT! Check to be sure you have simplified completely:

5. Simplifying Square Roots: An Alternate Method NOTE: If you have a perfect square (or perfect square factor) remaining under the radical symbol, you have not simplified completely.

6. Simplifying Square Roots KEY: L K for perfect squares or perfect square factors. NOTE: If you have a perfect square (or perfect square factor) remaining under the radical symbol, you have not simplified completely.

7. Simplifying Radicals: Part II  To multiply, then simplify square roots when possible

8. Product of Square Roots PRODUCT OF SQUARE ROOTS For all real numbers x ≥ 0, y ≥ 0, √x ∙√x = √x 2 = x √x ∙√y = √x∙y NOTE: Squaring a number and finding the square root are inverse operations.

9. Multiplying Square Roots with Common Radicands √3 ∙√3 =(√3) 2 =__ √4 ∙√4 =____ =__ √5 ∙√5 =____ =__ (2√3) 2 =_ ∙ _ =__ (3√5) 2 =_ ∙ _ =__ (2√5) 2 =_ ∙ _ =__ 3 (√4) 2 4 (√5) 2 5 4 3 12 9 5 45 4 5 20 NOTE: Squaring a number and finding the square root are inverse operations.

10. Multiplying Square Roots with Different Radicands √3 ∙√6 =√18 =____ ________ √2 ∙√10 =____ =____________ √4 ∙√20 =____ =____________ (2√3) (5√3) =____ =____________ (3√2) (5√2) =____ =____________ (3√2) (2√6) =____ =____________ (5√3) (√6) =____ =____________ √9 ∙2= 3√ 2 √20 √4 ∙5 = 4√ 5 √80 √16 ∙5 = 2√ 5 10 ∙3 30 6 ∙2 12 6 ∙ √12 5 ∙ √18

11. Dividing Radicals  T T o simplify an expression containing a quotient of radicals

12. Quotient of Square Roots QUOTIENT OF SQUARE ROOTS For all real numbers x ≥ 0, y > 0

13. Simplify each expression: a. b.

14. Radical in Denominator Fraction Under √ Rationalizing Denominators: Rationalize 1 ALERT! When you rationalize, you are changing the form of the number, but not its value. Double Check: 1. Fraction under √ ? 2. Radical in Denominator?

15. More Quotients of Radicals

16. Summary A radical expression is in simplest form when  each radicand contains no factor, other than one, that is a perfect square  the denominator contains no radicals and  each radicand contains no fractions.

17. Final Checks for Understanding 1.Simplify: √3 ∙√12 2. Simplify: √2 ∙√32 3.Indicate why each expressions is not in simplest radical form. a.) 5x 2 b.)√8yc.) √3x 5y

18. Homework Assignments:  DAY 1: Simplifying Radicals WS  DAY 2: Multiplying and Dividing Radicals WS