University:
Carleton College
Course:
MATH 101 | Calculus with Problem Solving
Academic year:

2024
Views:

389

Pages:

5
Author:

Clint W.

CHAPTER 6 - RATIONAL EXPONENTS AND RADICAL FUNCTIONS Simplify 1. √121 11 2. √16x² 2x 3. ∛64 4 4. √16x²y⁴ 4xy² 5. √30x³y² * √6x³y⁵ √180x⁶y⁷ 6x³y⁴√5 6. 6√4a²b / √12a⁴b⁴ 6√(4a²b / 12a⁴b⁴) 6√(1 / 3a²b³) 2ab√3 7. 5√12 5√(4*3) 10√3 8. (6√5)(2√6) 12√30 9. √15 / √5 √(15/5) √3 10. 6√2 + 3√2 - 7 9√2 - 7 11. 3√16 - 4√75 + √3 3*4 - 4*5√3 + √3 12 - 20√3 + √3 12 - 19√3 12. (4+√6)(4-√6) 16 - √36 16 - 6 10 Rationalize the denominator. 13. 2/√6 (2/√6)(√6/√6) 2√6 / 6 √6 / 3 14. 5 / √2 (5/√2)(√2/√2) 5√2 / 2 15. 3 / (1+√2) (3/(1+√2))(1-√2)/(1-√2) 3 - 3√2 / 1 - 2 -3 + 3√2 Express the following using rational exponents. 16. ⁴√15 15^(1/4) 17. ∛x² x^(2/3) 18. ⁷√(3x²y⁴) (3x²y⁴)^(1/7) 3^(1/7)x^(2/7)y^(4/7) Express the following in simplest radical form. No decimal answers. 19. 5√r³s⁵ 5r^(3/2)s^(5/2) 20. √(x³y²) x^(3/2)y 21. ∛54 ∛(27 * 2) 3∛2 22. ∛(x⁶y⁹) x²y³ 23. (√5)⁴ 5² 25 24. (√2)⁴ - 2√4 4 - 2 * 2 0 25. √(2/4) √(1/2) 1/√2 √2/2 26. ⁴√7 * ⁷√7 7^(1/4) * 7^(1/7) 7^(11/28) 27. ∛√16 ∛4 √2 Solve Each Equation. 28. m⁴ - 625 = 0 m⁴ = 625 m = ±5, ±5i 29. (x+4)³ = -33 x + 4 = ∛(-33) x = -4 - ∛33 ≈ -7.208 30. t⁴ - 21t² + 80 = 0 (t² - 16)(t² - 5) = 0 (t - 4)(t + 4)(t² - 5) = 0 t = 4, -4, ±√5 Solve the following equations. 31. √(x-3) = 7 x - 3 = 49 x = 52 32. ∛(2a+4) = 2 2a + 4 = 8 2a = 4 a = 2 33. √(8x+9) + 3 = 6 √(8x+9) = 3 8x + 9 = 9 8x = 0 x = 0 34. √x + √(x+9) = 3 √(x+9) = 3 - √x x + 9 = 9 - 6√x + x 6√x = 0 √x = 0 x = 0 35. √(7x-7) = √(3x-2) 7x - 7 = 3x - 2 4x = 5 x = 5/4 36. (x-6)² = √3x x² - 12x + 36 = 3√x x² - 15x + 36 = 0 (x - 12)(x - 3) = 0 x = 12, x = 3 37. (x-2)^(2/3) = 16 x - 2 = 16^(3/2) x - 2 = 8³ x - 2 = 512 x = 514 38. (3x - 5)^(5/3) = 243 3x - 5 = (243)^(3/5) 3x - 5 = 3³ 3x - 5 = 27 3x = 32 x = 32/3 39. (7x + 8)^(3/2) = 54 (7x + 8)^(3/2) = (216)^(1/2) 7x + 8 = (216)^(2/3) 7x + 8 = 6² 7x + 8 = 36 7x = 28 x = 4 Find the inverse of each function. Then graph the function and its inverse. 40. g(x) = 2x - 1 x = 2y - 1 x + 1 = 2y y = (x+1)/2 g⁻¹(x) = (x+1)/2 [Image of the graph of g(x) = 2x - 1 and its inverse g⁻¹(x) = (x+1)/2] 41. f(x) = 3x x = 3y y = x/3 f⁻¹(x) = x/3 [Image of the graph of f(x) = 3x and its inverse f⁻¹(x) = x/3] 42. y = 2/3x + 2 x = 2/3y + 2 3/2(x - 2) = y y⁻¹ = 3/2x - 3 [Image of the graph of y = 2/3x + 2 and its inverse y⁻¹ = 3/2x - 3] Determine whether each pair of functions are inverse functions. 43. f(x) = 5x - 5 g(x) = 1/5x + 1 f(g(x)) = 5(1/5x + 1) - 5 = x + 5 - 5 = x g(f(x)) = 1/5(5x - 5) + 1 = x - 1 + 1 = x Yes 44. f(x) = 6x - 2 g(x) = 1/6x + 3 f(g(x)) = 6(1/6x + 3) - 2 = x + 18 - 2 = x + 16 No