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 Radical Expressions

2. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §6.8 → Direct/Indirect Variation & Modeling  Any QUESTIONS About HomeWork §6.8 → HW-23 6.8 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 3 Bruce Mayer, PE Chabot College Mathematics Square Root The number c is a square root of a if c 2 = a

4. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 4 Bruce Mayer, PE Chabot College Mathematics Square Root Examples  Find the square roots: a) 144 b) 625  Solution a) The square roots of 144 are 12 and −12. To check, note that 12 2 = 144 and (−12) 2 = (−12)(−12) = 144  Solution b) The square roots of 625 are 25 and −25. To check, note that 25 2 = 625 and (−25) 2 = (−25)(−25) = 625.

5. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 5 Bruce Mayer, PE Chabot College Mathematics Notation & Nomenclature Notes  The NONnegative square root of a number is called the PRINCIPAL square root of that number.  A radical sign, √, indicates the principal square root of the number under the sign (the radicand).

6. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 6 Bruce Mayer, PE Chabot College Mathematics Examples  Principle Sq Roots  Find the following: a) b)  Soln a) The principal square root of 100 is its positive square root, so  Soln b) The symbol represents the opposite of  Since we have

7. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 7 Bruce Mayer, PE Chabot College Mathematics Expressions of the Form  It is tempting to write but the next example shows that, as a rule, this is UNtrue.  Example 

8. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 8 Bruce Mayer, PE Chabot College Mathematics Simplifying  For any real number a,  That is, The principal square root of a 2 is the absolute value of a.

9. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  SqRt & AbsVal  Simplify each expression. Assume that the variable can represent any real no.  SOLUTION Since y + 3 might be negative, absolute-value notation is necessary.

10. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  SqRt & AbsVal  SOLUTION b) & c)  SOLUTION b) Note that (m 6 ) 2 = m 12 and that m 6 is NEVER negative. Thus,  SOLUTION c) Note that (x 5 ) 2 = x 10 and that x 5 MIGHT be negative. Thus

11. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 11 Bruce Mayer, PE Chabot College Mathematics Plot f(x) = SqRt(x) = √(x)  Plot Using T-Table  Plot Pts and Connect with Smooth Curve

12. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 12 Bruce Mayer, PE Chabot College Mathematics Domain & Range of √x  Recall that taking the Sq-Root of a Negative Number does NOT return a Real-Number Result. Thus the Domain: {x|x≥0}  Recall the PRINCIPAL Sq-Root function return the POSITIVE Root only Thus the Range for the Principal SqRt fcn: {y|y≥0}

13. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 13 Bruce Mayer, PE Chabot College Mathematics Domain & Range of √x  Find Domain & Range for √x (the principal SqRt Fcn) from the Graph  Analysis of the Graph Reveals Domain = {x|x≥0} & Range = {y|y≥0}  Thus the SqRt Fcn occupies only the 1 st Quadrant of the XY Plane

14. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 14 Bruce Mayer, PE Chabot College Mathematics Domain & Range for  Need POSITIVE Radicand thus need x ≥ −3  Also the output of the principal SqRt Fcn is Always NONnegative so y is at MINIMUM −5  Thus Domain = (−3,  ) Range = (−5,  )  Graph Confirms D & R

15. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 15 Bruce Mayer, PE Chabot College Mathematics Radicands & Radical Expressions  A radical expression is an algebraic expression that contains at least one radical sign  Some examples:

16. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 16 Bruce Mayer, PE Chabot College Mathematics Examples  Radicands  Identify the radicand in expressions: a)b)  Soln a) in the radicand is y  Soln b) in the radicand is y 2 – 6

17. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 17 Bruce Mayer, PE Chabot College Mathematics Example  Elliptical Orbit  When calculating the velocity of a body in elliptical orbit at a distance r from the focus, in terms of the SemiMajor axis, a, we encounter the Expression:  Evaluate for: r = 10 260a = 14 460

18. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 18 Bruce Mayer, PE Chabot College Mathematics Square Root Functions  Given a PolyNomial, P, then a Square-Root Function takes the form  EXAMPLE  find f(3) for  SOLUTION: To find f(3), substitute 3 for x and simplify.

19. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 19 Bruce Mayer, PE Chabot College Mathematics Domain of Square Root Fcn  EXAMPLE  Find the Domain for: a)b)  SOLUTION a) the radicand for a Sq-Root must be NONnegative thus  This InEquality requires this Domain:

20. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 20 Bruce Mayer, PE Chabot College Mathematics Domain of Square Root Fcn  EXAMPLE  Find the Domain for: a)b)  SOLUTION b) the radicand for a Sq-Root must be NONnegative thus  This InEquality requires this Domain:

21. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 21 Bruce Mayer, PE Chabot College Mathematics Industrial Engineering Modeling  The attendants at a downtown parking lot use staging-spaces to leave cars before they are taken to long-term parking stalls. The required number, N, of such spaces is approximated by the formula: where A is the average number of arrivals during peak hours.

22. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 22 Bruce Mayer, PE Chabot College Mathematics Industrial Engineering Modeling  For  Find the number of spaces needed when an average of 62 cars arrive during peak hours  SOLUTION → Substitute 62 into the formula and use a calculator to find an approximation: Note that we round up to 20 spaces because rounding down would create some overcrowding.

23. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 23 Bruce Mayer, PE Chabot College Mathematics Simplified Form of a Square Root  A radical expression for a square root is simplified when its radicand has no factor other than 1 that is a perfect square.

24. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Simplification  Simplify by factoring (note that all variables are assumed to represent nonnegative numbers). a)b)c)  Soln a)  Soln b)  Soln c)

25. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 25 Bruce Mayer, PE Chabot College Mathematics Example  Evaluation  Evaluate: for a = 4, b = 7 and c = −2.  Solution:

26. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 26 Bruce Mayer, PE Chabot College Mathematics Simplifying Sq-Roots of Powers  To take the square root of an EVEN power such as x 12, note that x 12 = (x 6 ) 2  Thus  The exponent of the square root is half the exponent of the radicand. That is,

27. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 27 Bruce Mayer, PE Chabot College Mathematics Example: Sq-Roots of Powers  Simplify: a)b)c)  Soln a) Half of 8 is 4.  Soln b) Half of 14 is 7.  Soln c) Half of 32 is 16.

28. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 28 Bruce Mayer, PE Chabot College Mathematics Example: More Powers  Simplify: a)b)  Solution a)  Solution b) Caution! The square root of x 16 is not x 4.

29. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 29 Bruce Mayer, PE Chabot College Mathematics Example  Free Fall  The time (t) it takes in seconds to fall d feet is given by  Familiarize: Need to find the time it takes for an object to fall 800 feet  Translate: Use the formula, substituting 800ft for d  CarryOut: Replace d with 800. Divide within the radical. Evaluate the square root.  Find the Free-Fall time for an 800ft Drop

30. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 30 Bruce Mayer, PE Chabot College Mathematics Example  Jump to HyperSpace  Your sister is 5 years older than you are. She decides she has had enough of Earth and needs a vacation. She takes a trip to the Omega-One star system. Her trip to Omega-One and back in a spacecraft traveling at an average speed v took 15 years, according to the clock and calendar on the spacecraft.

31. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 31 Bruce Mayer, PE Chabot College Mathematics Example  Jump to HyperSpace  (cont.) But on landing back on Earth, she discovers that her voyage took 25 years, according to the time on Earth. This means that, although you were 5 years younger than your sister before her vacation, you are now 5 years older than she is after the interstellar vacation!  Find the StarShip’s speed using Einstein’s time-dilation eqn:

32. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 32 Bruce Mayer, PE Chabot College Mathematics Example  Jump to HyperSpace  SOLN: Sub t 0 = 15 (moving-frame time) and t = 25 (fixed-frame time) to obtain  So the StarShip was moving at 80% the speed of light (0.8c)

33. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 33 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §7.1 Exercise Set 16, 18, 24, 46, 100  Twins Encounter Time Dilation

34. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 34 Bruce Mayer, PE Chabot College Mathematics All Done for Today Child BMI Growth Chart

35. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 35 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

36. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 36 Bruce Mayer, PE Chabot College Mathematics All Done for Today SkidMark Analysis Skid Distances

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

38. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 38 Bruce Mayer, PE Chabot College Mathematics

39. BMayer@ChabotCollege.edu MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt 39 Bruce Mayer, PE Chabot College Mathematics