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1 Algebra II β Chapter 6 Day #1 Topic: Roots and Radical Expressions Standards/Goals: οΌ A.SSE.2: I can take the structure of an expression and identify different ways to rewrite it οΌ G.1.b.: I can simplify radicals that have various indices. οΌ G.1.c.: I can use properties of roots and rational exponents to simplify expressions. Today, we will focus on introducing ourselves to some new aspects to square roots and radicals that we have not encountered before. Here are some quiz things to consider when considering a radical: π βπ οΆ β is known as the βradical signβ οΆ βaβ is known as the βradicandβ οΆ βnβ is known as the βindexβ OR βrootβ o Some things to know about: ππ = π ο§ If βnβ is odd, then there is ONE real root ο§ If βnβ is even and positive, then there are TWO real roots. The positive root is what we call the βprincipalβ root. ο§ If βnβ is even and negative, then there are NO real roots. ο§ The only βnthβ root of 0 is 0. We first want to be able find all the βrealβ roots of a number. EXAMPLES: What are the real CUBE roots of the following numbers? 27 #1. -64 #2. 0.001 #3. 125 What are all the real square roots of the following numbers? 25 #4. -0.81 #5. #6. 121 What are the real fourth roots of the following numbers? #7. 1 #8. -0.00001 #9. What are the real fifth roots of the following numbers? #10. 0 #11. -1 #12. 32 49 16 81 2 In this chapter, we are only focusing on βrealβ roots. In the past, we focused on imaginary and complex roots. According to the Fundamental Theorem of Algebra, π₯ 4 β 1 = 0, has only four roots, only two of which are real. In a problem like this, letβs only focus on the real roots only. We now want to find the real roots of numbers. What is each real number root of the following numbers? 3 #3. β(β5)2 3 #6. β(β17)2 4 #9. β0.04 #1. β0.16 #2. ββ343 #4. β0.36 #5. ββ1 3 #8. ββ1 #7. ββ8 We also want to be able to simplify radical expressions when they have variables present. Before, we learn how to simplify radical expressions, letβs go over an important property that is important to consider. PROPERTY: for βnthβ Roots of βnthβ Powers π, ππ π ππ πππ π For any real numbers a, βππ = { |π| ππ π ππ ππ£ππ What is the simplified form of each radical expression? 4 #3. β16π₯ 8 3 #6. βπ₯ 8 π¦12 3 #9. βπ₯12 π¦16 #1. β25π4 #2. βπ12 π 4 #4. β9π₯ 2 #5. βπ₯ 6 π¦15 3 #8. βπ12 π15 #7. βπ6 π 9 4 4 3 HOMEWORK β Chapter 6 Day #1 Name ________________________________ Date _____ Find all the real square roots of each number. 1. 400 2. ο196 3. 10,000 4. 0.0625 Find all the real cube roots of each number. 6. ο343 5. 216 7. ο0.064 8. 1000 27 Find all the real fourth roots of each number. 9. ο81 10. 256 11. 0.0001 12. 625 Find each real root. 13. 144 14. ο 25 16. 3 0.001 17. 4 0.0081 18. 3 15. ο0.01 19. 27 3 ο27 20. 0.09 Simplify each radical expression. Use absolute value symbols when needed. 81x 4 21. 25. 5 22. 243x5 y15 121y10 26. 3 23. ο0.008 3 8g 6 24. 4 30. x4 81 3 125x 9 31. 36x 2 y 6