GP1-‐CA-‐S1-‐HW7 Last name: _______________________________ Period: _____ Seat #: _______
Test on Thursday August 20 Skill 1 Review
Selected Solutions Simplify the following radicals using the imaginary unit 1. −72 1. 6𝑖 2
Solution −72
= 𝑖 36 ∙ 2
= 6𝑖 2
Evaluate:
4. 𝑖!" 4. −𝑖
Solution 15÷ 4 = 3.75 The remainder is 3, therefore 𝑖!" = −𝑖 5. 𝑖!! 5. −1
Solution 15÷ 4 = 16.5 The remainder is 2, therefore 𝑖!! = −1 9. 12− 5𝑖 − 16+ 7𝑖 + (1+ 2𝑖) 9. −3− 10𝑖
Solution = 12− 5𝑖 − 16− 7𝑖 + 1+ 2𝑖
= 12− 16+ 1− 5𝑖 − 7𝑖 + 2𝑖
= −3− 10𝑖 11. −3𝑖(7− 8𝑖) 11. −24− 21𝑖
Solution = −21𝑖 + 24𝑖!
= −21𝑖 + 24 −1
= −24− 21𝑖
GP1-‐CA-‐S1-‐HW7 Last name: _______________________________ Period: _____ Seat #: _______ 13. (6+ 2𝑖)(−1+ 𝑖)(3− 2𝑖) 13. −16+ 28𝑖
Solution −1 𝑖 6 −6 6𝑖 2𝑖 −2𝑖 2𝑖! 6+ 2𝑖 −1+ 𝑖 = −8+ 4𝑖 So, −8+ 4𝑖 3− 2𝑖 3 −2𝑖 −8 −24 16𝑖 4𝑖 12𝑖 −8𝑖! −8+ 4𝑖 3− 2𝑖 = −16+ 28𝑖 14. 6 3− 5𝑖 − 8𝑖(2− 𝑖) 14. 10− 46𝑖 Solution = 18− 30𝑖 − 16𝑖 + 8𝑖! = 18− 8− 30𝑖 − 16𝑖 = 10− 46𝑖
18. Multiply the complex number −5− 8𝑖 by the complex conjugate. 18. 89
Solution −5− 8𝑖 −5+ 8𝑖 = −5 ! + 8! = 89
GP1-‐CA-‐S1-‐HW7 Last name: _______________________________ Period: _____ Seat #: _______ Evaluate:
19. (1− 3𝑖)! 19. −26+ 18𝑖
Solution 1− 3𝑖 1− 3𝑖 1 −3𝑖 1 1 −3𝑖 −3𝑖 −3𝑖 9𝑖! 1− 3𝑖 1− 3𝑖 = −8− 6𝑖 So, 1− 3𝑖 ! = 1− 3𝑖 1− 3𝑖 1− 3𝑖 = −8− 6𝑖 1− 3𝑖 1 −3𝑖 −8 −8 24𝑖 −6𝑖 −6𝑖 18𝑖! = −26+ 18𝑖
20. (7+ 2𝑖)(2− 5𝑖)(2+ 5𝑖)(7− 2𝑖) 20. 1537
Solution = 7+ 2𝑖 7− 2𝑖 2− 5𝑖 2+ 5𝑖 = 7! + 2! 2! + 5! = 53 29 = 1537
GP1-‐CA-‐S1-‐HW7 Last name: _______________________________ Period: _____ Seat #: _______ 23. !!!
!"!!! 23. !
!
Solution
=7− 1𝑖14− 2𝑖
14+ 2𝑖14+ 2𝑖
14 2𝑖 7 98 14𝑖 −1𝑖 −14𝑖 −2𝑖!
=100
14! + 2!
=12
25. !!!!!
25. !"!"+ !"
!"𝑖
Solution
=7
3− 2𝑖3+ 2𝑖3+ 2𝑖
=21+ 14𝑖3! + 2!
=2113+
1413 𝑖
GP1-‐CA-‐S1-‐HW7 Last name: _______________________________ Period: _____ Seat #: _______ Evaluate:
26. !!!!!!!
+ !!!!!!!
26. !"!"+ !!"
!"𝑖
Solution
=6− 4𝑖3+ 1𝑖
3− 1𝑖3− 1𝑖 +
8− 1𝑖1+ 3𝑖
1− 3𝑖1− 3𝑖
3 −1𝑖 1 −3𝑖 6 18 −6𝑖 8 8 −24𝑖 −4𝑖 −12𝑖 4𝑖! −1𝑖 −1𝑖 3𝑖!
=14− 18𝑖3! + 1! +
5− 25𝑖1! + 3!
=19− 43𝑖
10
=1910+
−4310 𝑖
27. !!!!!!!
− !!!!!!!!
27. !"!"+ !"
!"𝑖
Solution
=8− 1𝑖 − 6− 7𝑖
4+ 3𝑖
=2+ 6𝑖4+ 3𝑖
4− 3𝑖4− 3𝑖
4 −3𝑖 2 8 −6𝑖 6𝑖 24𝑖 −18𝑖!
=26+ 18𝑖4! + 3!
=26+ 18𝑖
25
=2625+
1825 𝑖
GP1-‐CA-‐S1-‐HW7 Last name: _______________________________ Period: _____ Seat #: _______ 29. !"!!"!
!!!! 29. 26
Solution 16+ 28𝑖6− 2𝑖
6+ 2𝑖6+ 2𝑖
6 2𝑖 16 96 32𝑖 28𝑖 168𝑖 56𝑖!
=40+ 200𝑖6! + 2!
=4040+
20040 𝑖
= 1+ 5𝑖 So, !"!!"!
!!!!= 1+ 5𝑖 = 1! + 5! = 26
30. −15+ 8𝑖 30. 17
Solution = −15 ! + 8! = 289 = 17 31. (9− 2𝑖)(4+ 9𝑖) 31. 8245
Solution 4 9𝑖 9 36 81𝑖 −2𝑖 −8𝑖 −18𝑖! = 54+ 73𝑖 = 54! + 73! = 8245 Note: 8245 = 5 ∙ 17 ∙ 97
GP1-‐CA-‐S1-‐HW7 Last name: _______________________________ Period: _____ Seat #: _______
Plot each complex number in the complex plane:
32. −1 + 𝑖
Plot −1, 1
34. − 4𝑖
Plot 0,−4