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Please complete the prerequisite Skills PG 412 #1-12

Please complete the prerequisite Skills PG 412 #1-12. Chapter 6: Rational Exponents and Radical Functions. Big ideas: Use Rational Exponents Performing function operations and finding inverse functions Solving radical equations. Lesson 1: Evaluate nth Roots and Use Rational Exponents.

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Please complete the prerequisite Skills PG 412 #1-12

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  1. Please complete the prerequisite SkillsPG 412 #1-12

  2. Chapter 6:Rational Exponents and Radical Functions Big ideas: Use Rational Exponents Performing function operations and finding inverse functions Solving radical equations

  3. Lesson 1: Evaluate nth Roots and Use Rational Exponents

  4. Essential question What is the relationship between nth roots and rational exponents?

  5. VOCABULARY • Nth root of a: For an integer n greater than 1, if bn = a, then b is an nth root of a. written as • Index of a radical: The integer n, greater than 1, in the expression

  6. a. Because n = 3 is odd and a = –216 < 0, –216 has one real cube root. Because (–6)3= –216, you can write = 3√–216 = –6 or (–216)1/3 = –6. b. Because n = 4 is even and a = 81 > 0, 81 has two real fourth roots. Because 34 = 81 and (–3)4 = 81, you can write ±4√ 81 =±3 EXAMPLE 1 Find nth roots Find the indicated real nth root(s) of a. a. n = 3, a = –216 b. n = 4, a = 81 SOLUTION

  7. 1 1 23 323/5 64 ( )3 = (161/2)3 = 43 = 43 = 64 = = 16  1 1 1 1 1 1 = = = = ( )3 (321/5)3 323/5 32  5 8 23 8 = = = = EXAMPLE 2 Evaluate expressions with rational exponents Evaluate (a) 163/2 and (b)32–3/5. SOLUTION Radical Form Rational Exponent Form a. 163/2 163/2 b. 32–3/5 32–3/5

  8. Keystrokes Expression Display 9 1 5 7 3 4 12 3 8 7  c. ( 4 )3 = 73/4 EXAMPLE 3 Approximate roots with a calculator a. 91/5 1.551845574 b. 123/8 2.539176951 4.303517071

  9. for Examples 1, 2 and 3 GUIDED PRACTICE Find the indicated real nth root(s) of a. 1. n = 4, a = 625 3. n = 3, a = –64. SOLUTION ±5 SOLUTION –4 2.n = 6, a = 64 4. n = 5, a = 243 SOLUTION ±2 SOLUTION 3

  10. 1 3 for Examples 1, 2 and 3 GUIDED PRACTICE Evaluate expressions without using a calculator. 5. 45/2 7. 813/4 27 SOLUTION 32 SOLUTION 6. 9–1/2 8. 17/8 SOLUTION SOLUTION 1

  11. Expression 10. 64 2/3 – 11. (4√ 16)5 12. (3√–30)2 for Examples 1, 2 and 3 GUIDED PRACTICE Evaluate the expression using a calculator. Round the result to two decimal places when appropriate. 9. 42/5 1.74 SOLUTION SOLUTION 0.06 SOLUTION 32 9.65 SOLUTION

  12. a. 4x5 = 128 x5 = 32 x = 32  5 x 2 = EXAMPLE 4 Solve equations using nth roots Solve the equation. Divide each side by 4. Take fifth root of each side. Simplify.

  13. b. (x – 3)4 = 21 + x – 3 = 21 – 4  + x = 21 + 3 – 4  or 21 + 3 x = x = – 21 + 3 4  4  5.14 0.86 x or x EXAMPLE 4 Solve equations using nth roots Take fourth roots of each side. Add 3 to each side. Write solutions separately. Use a calculator.

  14. Essential question The nth root of a can be written as a to the What is the relationship between nth roots and rational exponents?

  15. Simplify the expression:43*48

  16. Lesson 2: Apply Properties of rational exponents

  17. Essential question How are the properties of rational exponents related to properties of integer exponents?

  18. VOCABULARY • Simplest form of a radical: A radical with index n is in simplest form if the radicand has no perfect nth powers as factors and any denominator has been rationalized • Like radicals: Radical expressions with the same index and radicand

  19. 51 5 51/3 51/3 a. 71/4 71/2 b. (61/2 41/3)2 = (61/2)2 (41/3)2 = 6(1/22) 4(1/32) = 6 42/3 = 61 42/3 1 c. (45 35)–1/5 = [(4 3)5]–1/5 = 12[5 (–1/5)] = 12 d. = 2 42 421/3 2 1/3 e. = = 7(1/3 2) 6 61/3 EXAMPLE 1 Use properties of exponents Use the properties of rational exponents to simplify the expression. = 73/4 = 7(1/4 + 1/2) = 12 –1 = (125)–1/5 = 5(1 – 1/3) = 52/3 = 72/3 = (71/3)2

  20. a. 5 80 12 16 18 216 = 12 18 = = 6 4 4 4 3 3 3 3 80 b. = = = 2 4 5 EXAMPLE 3 Use properties of radicals Use the properties of radicals to simplify the expression. Product property Quotient property

  21.  5 5 3 3 3  3 27  5 27 a. = = 3 = 3  135 EXAMPLE 4 Write radicals in simplest form Write the expression in simplest form. Factor out perfect cube. Product property Simplify.

  22.      7 7 8 4 8 4 5 5 5 5 5 5    5 5 5 28 28 32 = b. = = 2 EXAMPLE 4 Write radicals in simplest form Make denominator a perfect fifth power. Product property Simplify.

  23.     2 2 2 2 2 3 3 3 3 3 (1 + 7) a. 8 7 + = = 4 3 4 4 4 3       3 10 10 54 27 10 10 b. = = + (81/5) 2 – 3 (3 – 1)  – – c.  2 3 = = = 2 (81/5) (81/5) 12 10 2 = (2 +10) (81/5) EXAMPLE 5 Add and subtract like radicals and roots Simplify the expression.

  24. 3   5 3 24 2   3 250 + 40 3 5 4 4 4 27 3 3 3 3   5 5 3 5 2 for Examples 3, 4, and 5 GUIDED PRACTICE Simplify the expression. SOLUTION SOLUTION SOLUTION SOLUTION

  25. 3  43(y2)3 a. 4y2 = = = 3pq4 b. (27p3q12)1/3 271/3(p3)1/3(q12)1/3 = = = 3 4 4 4 3p(3 1/3)q(12 1/3)     n8 43 m4 m4 m 3 3   (y2)3 64y6 14xy 1/3 4 7x1/4y1/3z6 7x(1 – 3/4)y1/3z –(–6) = = c. = = = n2 2x 3/4 z –6 4  (n2)4 d. m4 n8 EXAMPLE 6 Simplify expressions involving variables Simplify the expression. Assume all variables are positive.

  26. 5  = 5 4a8b14c5  4a5a3b10b4c5 5 5   a5b10c5 4a3b4 a. = = b. = x 5 ab2c  4a3b4 y8 x y 3 x y 3 = 3 y9 y8 y EXAMPLE 7 Write variable expressions in simplest form Write the expression in simplest form. Assume all variables are positive. Factor out perfect fifth powers. Product property Simplify. Make denominator a perfect cube. Simplify.

  27. 3  xy = 3  x y 3 =  y9 y3 EXAMPLE 7 Write variable expressions in simplest form Quotient property Simplify.

  28. 3z) (12z – 1 3 w w   + + 5 5 a. = = 9z 3 3   2z2 2z5 3 1 4 (3 – 8) xy1/4 –5xy1/4 b. = = – 3xy1/4 8xy1/4 5 5 5 c. = z = 12 – w w   = 3 3 3 3 3z     – 2z2 2z2 54z2 2z2 12z EXAMPLE 8 Add and subtract expressions involving variables Perform the indicated operation. Assume all variables are positive.

  29. 6xy 3/4 3x 1/2 y 1/2 3q3 3  27q9 5 2x1/2y1/4 –  w  w3 9w5 x10 y5 x2 y w 2w2  for Examples 6, 7, and 8 GUIDED PRACTICE Simplify the expression. Assume all variables are positive. SOLUTION SOLUTION SOLUTION SOLUTION

  30. Essential question All properties of integer exponents also apply to rational exponents How are the properties of rational exponents related to properties of integer exponents?

  31. Let f(x) = 3x + 5. Find f(-6)

  32. Lesson 3Perform Function operations and composition

  33. Essential question What operations can be performed on a pair of functions to obtain a third function?

  34. VOCABULARY • Power Function: A function of the form y=axb, where a is a real number and b is a rational number • Composition: The composition of a function g with a function f is h(x) = f(f(x)).

  35. a. f(x) + g(x) f(x) – g(x) b. EXAMPLE 1 Add and subtract functions Letf (x)= 4x1/2andg(x)=–9x1/2. Find the following. SOLUTION f (x) + g(x) = 4x1/2 + (–9x1/2) = [4 + (–9)]x1/2 = –5x1/2 SOLUTION f (x) – g(x) = [4 – (–9)]x1/2 = 13x1/2 = 4x1/2 – (–9x1/2)

  36. The functions fand geach have the same domain: all nonnegative real numbers. So, the domains of f + gand f – galso consist of all nonnegative real numbers. c. the domains of f + gand f – g EXAMPLE 1 Add and subtract functions SOLUTION

  37. a. f (x) g(x) b. 6x f (x) f (x) g(x) g(x) x3/4 f (x) g(x) = = 6x1/4 = 6x(1 – 3/4) EXAMPLE 2 Multiply and divide functions Let f (x)= 6xand g(x) = x3/4. Find the following. SOLUTION = (6x)(x3/4) = 6x(1 + 3/4) = 6x7/4 SOLUTION

  38. The domain of f consists of all real numbers, and the domain of gconsists of all nonnegative real numbers. So, the domain of f gconsists of all nonnegative real numbers. Because g(0) = 0, the domain of is restricted to all positive real numbers. f g EXAMPLE 2 Multiply and divide functions f g and the domains of f c. g SOLUTION

  39. r(m) s(m) (6 106)m0.2 = 241m–0.25 = • Findr(m) s(m). EXAMPLE 3 Solve a multi-step problem Rhinos For a white rhino, heart rate r(in beats per minute) and life span s(in minutes) are related to body mass m(in kilograms) by these functions: • Explain what this product represents.

  40. Find and simplify r(m) s(m). r(m) s(m) = 241m –0.25 [ (6 106)m0.2 ] 241(6 106)m(–0.25 + 0.2) = (1446 106)m –0.05 = (1.446 109)m –0.05 = EXAMPLE 3 Solve a multi-step problem SOLUTION STEP 1 Write product of r(m) and s(m). Product of powers property Simplify. Use scientific notation.

  41. Interpret r(m) s(m). EXAMPLE 3 Solve a multi-step problem STEP 2 Multiplying heart rate by life span gives the total number of heartbeats for a white rhino over its entire lifetime.

  42. f (x) + g(x) f (x) – g(x) for Examples 1, 2, and 3 GUIDED PRACTICE Let f (x) = –2x2/3andg(x) = 7x2/3. Find the following. SOLUTION f (x) + g(x) = –2x2/3 + 7x2/3 = 5x2/3 = (–2 + 7)x2/3 SOLUTION f (x) – g(x) = –2x2/3 – 7x2/3 = [–2 + ( –7)]x2/3 = –9x2/3

  43. the domains of f + gand f – g for Examples 1, 2, and 3 GUIDED PRACTICE SOLUTION all real numbers; all real numbers

  44. f (x) g(x) f (x) g(x) for Examples 1, 2, and 3 GUIDED PRACTICE Let f (x) = 3xandg(x) = x1/5. Find the following. SOLUTION 3x6/5 SOLUTION 3x4/5

  45. the domains off g and f g for Examples 1, 2, and 3 GUIDED PRACTICE SOLUTION all real numbers; all real numbers except x=0.

  46. Use the result of Example 3 to find a white rhino’s number of heartbeats over its lifetime if its body mass is 1.7 105kilograms. about 7.92 108 heartbeats for Examples 1, 2, and 3 GUIDED PRACTICE Rhinos SOLUTION

  47. Essential question Two functions can be combined by the operations: +, -, x, ÷ and composition What operations can be performed on a pair of functions to obtain a third function?

  48. Solve x=4y3 for y

  49. Lesson 4: Use inverse Functions

  50. Essential question How do you find an inverse relation of a given function?

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