1. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §7.1 Cube & nth Roots

2. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §7.1 → Square-Roots and Radical Expressions  Any QUESTIONS About HomeWork §7.1 → HW-24 7.1 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 3 Bruce Mayer, PE Chabot College Mathematics Cube Root  The CUBE root, c, of a Number a is written as:  The number c is the cube root of a, if the third power of c is a; that is; if c 3 = a, then

4. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 4 Bruce Mayer, PE Chabot College Mathematics Example  Cube Root of No.s  Find Cube Roots a) b) c)  SOLUTION a) As 0.2·0.2·0.2 = 0.008 b) As (−13)(−13)(−13) = −2197 c) As 3 3 = 27 and 4 3 = 64, so (3/4) 3 = 27/64

5. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 5 Bruce Mayer, PE Chabot College Mathematics Cube Root Functions  Since EVERY Real Number has a Cube Root Define the Cube Root Function:  The Graph Reveals Domain = {all Real numbers} Range = {all Real numbers}

6. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 6 Bruce Mayer, PE Chabot College Mathematics Evaluate Cube Root Functions  Evaluate Cube Root Functions a) b)  SOLUTION (using calculator) b)

7. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 7 Bruce Mayer, PE Chabot College Mathematics Simplify Cube Roots  For any Real Number, a  Use this property to simplify Cube Root Expressions.  For EXAMPLE  Simplify  SOLUTION because (–3x)(–3x)(–3x) = –27x 3

8. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 8 Bruce Mayer, PE Chabot College Mathematics n th Roots  nth root: The number c is an n th root of a number a if c n = a.  The fourth root of a number a is the number c for which c 4 = a. We write for the nth root. The number n is called the index (plural, indices). When the index is 2 (for a Square Root), the Index is ommitted.

9. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 9 Bruce Mayer, PE Chabot College Mathematics Odd & Even n th Roots →  When the index number, n, is ODD the root itself is also called ODD A Cube-Root (n = 3) is Odd. Other Odd roots share the properties of Cube-Roots – the most important property of ODD roots is that we can take the ODD-Root of any Real Number – positive or NEGATIVE –Domain of Odd Roots = (− , +  ) –Range of Odd Roots =(− , +  )

10. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  n th Roots of No.s  Find ODD Roots a)b) c)  SOLUTION a) Since 3 5 = 243 b) As (−3)(−3)(−3)(−3)(−3) = −243 c) When the index equals the exponent under the radical we recover the Base

11. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 11 Bruce Mayer, PE Chabot College Mathematics Odd & Even n th Roots →  When the index number, n, is EVEN the root itself is also called EVEN A Sq-Root (n = 2) is Even. Other Even roots share the properties of Sq-Roots –The most important property of EVEN roots is that we canNOT take the EVEN-Root of a NEGATIVE number. –Domain of Even Roots = {x|x ≥ 0} –Range of Even Roots = {y|y ≥ 0}

12. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  n th Roots of No.s  Find EVEN Roots a)b) c)  SOLUTION a) Since 3 4 = 81 b) Even Root is Not a Real No. c) Use absolute-value notation since m could represent a negative number

13. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 13 Bruce Mayer, PE Chabot College Mathematics Simplifying n th Roots na Even Positive a NegativeNot a real number |a||a| Odd Positive a Negative a

14. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Radical Expressions  Find n th Roots a) b) c)  SOLUTION a) b) c)

15. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 15 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §7.1 Exercise Set 50, 74, 84, 88, 98, 102  Principal n th Root

16. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 16 Bruce Mayer, PE Chabot College Mathematics All Done for Today SkidMark Analysis Skid Distances

17. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 17 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

18. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 18 Bruce Mayer, PE Chabot College Mathematics Graph y = |x|  Make T-table

19. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 19 Bruce Mayer, PE Chabot College Mathematics