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Algebra 1: Chapter 10 Notes 1 Algebra Homework: Chapter 10 (Homework is listed by date assigned; homework is due the following class period) Leave all answers in reduced, radical form. No decimals please!!! HW# 15 16 Date F 5/6 M 5/9 17 T 5/10 18 W 5/11 19 Th 5/12 20 F 5/13 21 M 5/16 In-Class Section 10.1: Simplifying and Multiplying Radicals Section 10.1 and 10.2: Dividing Radicals and the Pythagorean Theorem Section 10.3: Adding, Subtracting, and Distributing Radical Expressions Review of Sections 10.1-10.3 Section 10.4: Solving Radical Equations Chapter 10 Review Chapter 10 Test Homework HW#15 Pg. 489: 2-24 even Pg. 580: 41-48 all Get Progress Report Signed by Wednesday HW#16 Pg. 489: 15-23 odd, 28-50 even Pg. 496: 1-3, 7-9, 16-18 Get Progress Report Signed by Wednesday HW#17 Pg. 490: 33-51 odd Pg. 496: 10, 11, 20 Pg. 503: 1-3, 10-31x3 HW#18 Pg. 521: 10-48 even Pg. 580: 58-64 all HW#19 Pg. 506: 1-10 all Pg. 524: 1-8 all Pg. 510: 1-3, 9-12, 38, 45, 46 (Show ALL checks on pg. 510) Chapter 10 Test and Notes Check on Tuesday HW#20 Chapter 10 Study Guide Correct Study Guide Online Chapter 10 Test and Notes Check on Tuesday HW#21 Pg. 580: 49-52, 55, 65-72 Print: Chapter 9 Notes 1 Algebra 1: Chapter 10 Notes Notes #15: Real and Radical Expressions (Sections 10.1) A. Square Roots Complete: 2 02 = ____ 12 = ____ 22 = ____ 32 = ____ 42 = ____ 52 = ____ 62 = ____ 72 = ____ 82 = ____ 92 = ____ 102 = ____ 112 = ____ 122 = ____ 132 = ____ 142 = ____ 152 = ____ 202 = ____ 302 = ____ 402 = ____ 502 = ____ The numbers you just wrote down are called ___________ __________ because you _______________ another number to find them. Complete: What number squared makes 16? _____ or _____ What number squared makes 81? _____ or ______ What number squared makes 0? ______ What number squared makes -4? ______ You just found the ___________ _________ ( ) of each of these numbers. Positive numbers have ______ real square roots and negative numbers have _______ real square roots. How do you know which root to list, the positive, the negative, or both? 25 means the _____________ (or positive) square root of 25. 25 = ____ - 25 means the ______________ square root of 25. - 25 = ____ 25 means ___________ square roots of 25. 25 = ____ or ____ Find the square roots of each number: 1.) 81 2.) 64 3.) 100 Simplify: 4.) 9 5.) 8.) Evaluate 2 x 7 for x = 2. Is this a real number? 49 6.) 121 7.) 196 9.) Evaluate 4 8y for y = -1. Is this a real number? 2 Algebra 1: Chapter 10 Notes Simplify: (if it is a polynomial, factor first!!) 10.) 14.) m2 16b2 11.) x2 4 x 4 3 15.) 13.) 4d 2 12d 9 (2b 5) 2 16.) 18 B. Simplifying Square Roots For problems like #16, you use one of two methods: (1) Find perfect square factors of 18 to help you break it down (2) Write a complete factor tree for 18 and simplify by taking out “buddies” 18 18 Simplify each expression using BOTH methods. Then decide which you prefer: Assume that all variables are nonnegative. (If it is a polynomial - ____________ first!!) 1.) 24 2.) 4.) 48a 2 5.) 300 125b3 3.) 6.) 50m 42g 5 3 Algebra 1: Chapter 10 Notes x7 7.) 10.) 4 8.) 8( w 3) 4 (m 5)5 11.) 9.) 200(k 2)5 12.) 5x2 10 x 5 150a3b7 B. Multiplying Square Roots We will be multiplying expressions like: (2 Steps: - simplify each radical completely 20)(3 5) - (outside • outside) inside inside or - simplify your answer, if possible (a b )(c d ) ________ *Remember, if you are multiplying polynomials, you must ________** Multiply and simplify, if possible: 1.) 5 6 2.) 4.) 4 x 5.) 2 3 7 7 4 6 3.) 64 6.) 2 2 7 5 10 4 Algebra 1: Chapter 10 Notes 7.) 3 6a 3 10.) 7a 13.) 15c2d 3 4 8a 14b 20c3d 4 5 3 8.) 5 12 x y 11.) 14.) 3 2 -2 10 xy 5 2 2m2 n 9.) 3 6 x 2 3 3a3b2c 12.) 12a b c 10m3n 15.) 5 7 2 10 x 2 5 Algebra 1: Chapter 10 Notes 6 Notes #16: Dividing Radical Expressions and the Pythagorean Theorem (Sections 10.1 and 10.2) Section 10.1: Dividing Rational Expressions Taking the square root of a fraction is the same as taking the square root of the ___________________ and ______________________ separately OR You can ___________ first and then split up the fraction Examples: 9 16 12 27 Simplify: 1.) 4.) 81 100 6 24 2.) 121 196 3.) 5.) 75 300 20a 3 6.) 80a 8m3 7.) 2m5 3x3 25 144 60 y 7 24 x 4 8.) 9.) 3 y3 6 Algebra 1: Chapter 10 Notes 7 However, sometimes our fractions don’t simplify as well…we end up with a radical expression in the denominator. This is NOT allowed!! Ex: 24 30 Ex: 6x2 12 To fix this problem: - simplify the fraction as much as you can - multiply this simplified fraction (both ________ AND __________) by the exact term that is still in a square root sign on the denominator - simplify and reduce Simplify: 10.) 12.) 14.) 16.) 5 6 11.) 1 5 5 x 12 2 13.) 15.) 17.) 3 8 2c 6c3 3 2x d3 27 7 Algebra 1: Chapter 10 Notes 18.) 20.) 14 3x 8 19.) 18 18 x4 108 xy 5 2x 21.) y3 Section 10.2: The Pythagorean Theorem Solve for x: 1.) 32 + 42 = x2 2.) 132 = 122 + x2 3.) x2 + 42 = 82 Right Triangles: Triangles with one _____________ ______________. hypotenuse = ____________________ legs = __________________________ Pythagorean Theorem: ( _________)2 + (_________)2 = (___________________)2 OR a2 + b2 = c2 8 Algebra 1: Chapter 10 Notes 9 Use Pythagorean Theorem to solve for the third side of each right triangle. Leave your answer in simplified radical form: 4.) 5.) b 8 c 3 16 4 6.) 7.) x x x 2 3 6 4 8.) a = 5, b = 12, c = ? 9.) a = 1, c = 10.) A 15 foot ladder is leaning against a building. The bottom of the ladder is 9ft from the building. How high is the top of the ladder? 11.) How long must a wire be to reach from the top of a 12-m telephone pole to a point on the ground 5m from the foot of the pole? 3,b=? 9 Algebra 1: Chapter 10 Notes 10 Determine whether the given lengths can be sides of a right triangle. - use the longest length as c. Use the shorter two lengths as a and b - plug into the Pythagorean theorem: a2 + b2 = c2 - if both sides of the equation are equal, then the triangle is ________ If both sides of the equation are not equal, then the triangle is ______ _____________ 12.) 2ft, 3ft, 4ft 13.) 6in, 7in, 8in 14.) 5cm, 5cm, 5 2 cm Notes #17: Other Operations on Radical Expressions (Section 10.3) Adding and subtracting square roots is the same process as adding and subtracting ________________ ________________: look for ________________!! Review: Add/Subtract 1.) 3x – 2y – 8x + 7y 2.) -2mn – (-3x2) + mn – 7x2 Steps for adding/subtracting radical expressions: _______________ all radical expressions (break each term down as far as possible) Look for _________________ (underline, circle, box, etc) Combine like terms. Add the _______________, but leave the roots __________ Add/Subtract: 1.) 4.) 2 5 3 5 8 18 2.) 6 7 (3 7) 5.) 3.) 12 3 5 2 3 12 6 27 10 Algebra 1: Chapter 10 Notes 11 2 40 (3 90) 6.) 7 12 x 3 48x 8.) 2 10.) 13.) 2 3 4 6 7.) 9.) 2 2 3 4 5 3 11.) 2 3 14.) x 4x 3 12 4 20 3 45 (2 300) 3 5 8 6 5 6 5 5 4 5 12.) 13.) 3 2 1 2 What was special about #14? Steps for simplifying a fraction with a binomial in the denominator: Multiply the _______ and ____________ of the fraction by the conjugate 3 2 Ex: Ex: 2 5 3 5 Distribute in the numerator, use ________ in the denominator Reduce only if ________ terms can be simplified by the same factor 11 Algebra 1: Chapter 10 Notes 12 3 2 5 Ex: Ex: 2 3 5 2.) 6 10 2 4.) 15 8 5 Simplify 1.) 8 4 6 13 7 5 3.) Notes #18: Review of Sections 10.1 – 10.3 Simplify: 1.) 48a3b7 4.) 6 12( x 1) 2 6.) 5 2 2 12 2.) 5 x 2 y 24 x 3 y 4 3.) 4m2 n 63mn5 4 y 2 40 y 100 5.) 7.) 12 j 5 k 8 j 2 k 3 jk 2 20 jk 4 12 Algebra 1: Chapter 10 Notes 48m5 3m 8.) 9.) 11.) 3 2 6 2 14.) 1 7 16.) 13 8 2 10.) 12.) 5 6 7 5 3 19.) 4 5 125 40 2 13.) 1 3 2 4 3 15.) 5 12 2 17.) 37 3 2 18 x 2 5 18.) 12 5 27 20.) 3 18 6 72 13 Algebra 1: Chapter 10 Notes Notes #19 Review Concepts First: Solve for x 1.) x 2 42 52 2.) x 2 42 82 4.) x 12 5.) 14 3.) x 2 12 x 3 12 6.) x 3 12 Section 10.4: Solving Equations with radical expressions Steps: Get the all alone 2 Square ( ) both sides (If you square a binomial, be sure to use _______!!) Solve for x Plug it back to check for extraneous solutions (you can never take the square root of a ______________ number) Solve for x: 1.) 3.) x 8 2x 4 7 2.) x 1 3 4.) x 1 2x 5 14 Algebra 1: Chapter 10 Notes 15 5.) 3x 5 3 6.) 7.) x 2 5 3 8.) 3m 6 2m 3 10.) x 2 x2 8 9.) 4 10 5 x 11 x 1 x2 5 15 Algebra 1: Chapter 10 Notes Classwork Day #20: Chapter 10 Review Simplify each radical expression. 1. 56x8 y 7 5. 5 3 12. 2. 8a 2 108a9 2 9. 2 48 10 12 7 10 3 6 2 16 6. 7 3 6x 10. 6 5 7 3 10 13. 3. 81m8 100n16 4. 5ab 6a2b3 7. 12c 48b 4 8. 11. 6 2a 10ab 7 5 3 5 3 5 5 7 3 5 12 6 3 16 Algebra 1: Chapter 10 Notes Solve for the variable: 14. a2 + 62 = 122 17 15. 5 3 b 16. x2 108 2x 17. 3 6 x 2 17 Algebra 1: Chapter 10 Notes Algebra 1: Chapter 10 Study Guide 18 Name: __________________ Simplify each radical expression. 1. 48x 6 y 5 5. 49 y 8 64 x 6 6. 9. 48b 81 10. 2. 112m3n7 6 12 3 6 5 2 6m 3. 6a 72a5 7. 3 g 2 g 2 h 11. 56b 24c 4 4. 5d 2h 10 gh 8. 12. 121 64 7 5 2 5 11 6 11 18 Algebra 1: Chapter 10 Notes 19 14. 3 32 7 8 13. 2 28 6 63 Solve for the variable: 16. a2 + 42 = 52 17. 52 + b2 = 102 15. d3 27 18. 15 b 9 Simplify: 19. 5 2 7 2 6 22. 6 32 20. 4 32 2 5 3 2 23. 21. 5 15 3 2 3 6 3 7 19 Algebra 1: Chapter 10 Notes Solve each radical equation and check your answer(s) 24. x2 75 2 x 25. x 3 2 7 20 26. 5 7 x 5 20