Math Algebra Radicals Test Review Key

Algebra 2 Honors Radicals Test Review Name: Date: Determine whether the following statements are True or False. Be able to explain why. 1. √x³ = |x| False - odd roots don't need absolute value 2. √x⁵ = x True 3. ³√-x³ = -x True 4. √x² = x False - |x| 5. √x⁸ = x⁴ True 6. √(-x)² = -x False. √(-x)² = |x| Simplify. Assume that the given radicals denote real numbers. You may not leave radicals in the denominator or negative exponents. 1   1. mi01000971.schoolwires.net mi01000971.schoolwires.net 7. √x⁶ = |x³| 8. ³√81x¹³y⁸ = ³√3⁴x¹²y⁸ = 3x⁴y²³√3xy² 9. ⁴√625a¹⁶b⁶c² = ⁴√5⁴a¹⁶b⁶c² = 5a⁴|b|√bc 10. √-64x⁶y¹¹ = √-2⁶x⁶y¹¹ = -2x²y²√y³ = -2xy²√y 11. √27cd²√3cd² = √81c²d⁴ = 9cd² 12. ⁵√-16x⁵y³ = ⁵√-2⁵x⁵y³ = -xy⁵√y² = -xy√y 13. √(81x⁻⁵) / ³√(3x) = ³√27x⁻⁶ = 3/x² 14. √(48x⁵y) / ³√(3xy) = √16x⁴√3y / ³√3xy = 4x²√3y / ³√3xy = 4x² 15. √(y⁻²) / ³√(25y⁻⁵) = √(1/y²) / ³√(25/y⁵) = 1/y / (⁵√25 / ⁵√y⁵) = 1/y / (5/y) = 1/5 16. √(16x² - 25y²) = √(4x - 5y)(4x + 5y) 17. √(49a² + 56a + 16) = √(7a + 4)² = |7a + 4| 18. √112x⁴ - √7x⁴ = √2⁴ * 7x⁴ - x²√7 = 4x²√7 - x²√7 = 3x²√7 19. 3√20 + 5√45 + √75 = 3√(4*5) + 5√(9*5) + √(25*3) = 6√5 + 15√5 + 5√3 = 21√5 + 5√3 20. √18x⁶ + 5√8x⁶ = 3√2x³ + 10√2x³ = 13√2x³ 21. √2x³ + √50x³ - √8x³ = √2x³ + 5√2x³ - 2√2x³ = 4√2x³ 22. 5³√16x⁵y³ = 5√8x³y³ = 5 * 2xy√2y = 10xy√2y 23. (8¹/³)⁴ = (2³)⁴/³ = 2⁴ = 16 24. (2¹/²)⁴ = 2² = 4 25. √-81x⁶y¹¹ = √(-1 * 3⁴ * x⁶ * y¹⁰ * y) = -3x²y⁵√y 26. √(15x⁻²y¹/³)³ = (15³x⁻⁶y)¹/³ = 3√15x⁻²y¹/³ = 3√15y¹/³ / x² 27. √6³ * √36⁻² ÷ √36 = 6³/² * 36⁻¹ * 36⁻¹/² = 6³/² * 36⁻³/² = 6³/² / 36³/² = 1/6 28. √125x³y⁵ = 5xy²√5x 29. 10⁴/³ * 10¹/² * 10⁻⁴/³ = 10⁴/³⁺¹/²⁻⁴/³ = 10¹ = 10 30. (√81)⁵ * (√27)² = 3⁵ * 3³ = 3⁸ 31. √125 + √25⁻² = 5√5 + 1/5 32. (√√81)⁵ * (√√27)² = (3²)⁵/² * (3³/⁴)² = 3⁵ * 3³/² = 3²³⁄² 33. (10¹.²)² * √10².⁶ = 10².⁴ * 10¹³. = 10³⁷ 34. (2√5 + √2)(√5 - √3) = 10 - 2√15 + √10 - √6 35. (√3 + 2)⁻¹ = 1 / (√3 + 2) = (√3 - 2) / (√3 - 2)(√3 + 2) = (√3 - 2) / (3 - 4) = -√3 + 2 Simplify. You must rationalize all expressions. You may not leave negative exponents in your final answer. 36. (5 - √7)² / (5 + √7) = (25 - 10√7 + 7) / (5 + √7) = 32 - 10√7 / (5 + √7) * (5 - √7) / (5 - √7) = 160 - 32√7 - 50√7 + 70 / 25 - 7 = 230 - 82√7 / 18 = 115 - 41√7 / 9 37. (√a - √b) / (√a + √b) = (√a - √b) / (√a + √b) * (√a - √b) / (√a - √b) = a - 2√ab + b / a - b 38. 2 + √x + (1 - √x) / (4 - x) = 8 - 2x + 4√x - x√x + 14 - 7√x / 4 - x = 22 - 2x - 3√x - x√x / 4 - x Solve. If there are no solutions, so state. 39. 4a⁴(4a² - 3) = 3(27 - 4a²) 256a⁶ - 12a² = 81 - 12a² 256a⁶ = 81 a⁴ = 81/256 a = ±3/4 40. √3x - 5 = 5 √3x = 10 3x = 100 x = 100/3 41. √x³ - 5 = 5 √x³ = 10 x³ = 1000 x = 10 42. √(6b + 1) - 2 = 0 √(6b + 1) = 2 6b + 1 = 4 6b = 3 b = 1/2 43. √(d² - 19) - 2d + 11 = 0 √(d² - 19) = 2d - 11 d² - 19 = 4d² - 44d + 121 0 = 3d² - 44d + 140 0 = (3d - 14)(d - 10) d = 14/3, 10 44. u = 1/3√(6u - 1) 9u² = 6u - 1 9u² - 6u + 1 = 0 (3u - 1)² = 0 u = 1/3 45. 8f = 1 - 2√f (8f - 1)² = (-2√f)² 64f² - 16f + 1 = 4f 64f² - 20f + 1 = 0 (4f - 1)(16f - 1) = 0 f = 1/4, 1/16 46. 5(t - 3√t) + 3 = 3(t + 1) 5t - 15√t + 3 = 3t + 3 2t = 15√t 4t² = 225t 4t² - 225t = 0 t(4t - 225) = 0 t = 0, 225/4 47. √(2g² + 9) = √27 2g² + 9 = 27 2g² = 18 g² = 9 g = ±3