D.(1-i)(1+i)分之（1+2i)(1+3i) 5. (1-i)(1+i)分之2(1+2i)(1+3i)減5(2i三次方加i六次方)加(7i九次方加2i十二次方)=X(13+11i+12i二次方) 求X的值。?

D)
(1+2i)(1+3i)
-----------------
. . (1-i)(1+i)

. . 1+2i+3i+6i²
= ----------------
. . . 1-i+i - i²

. . 1+5i-6
= ----------
. . 1-(-1)

. . -5+5i
= ---------
. . . . 2

. . 5(-1+i)
= ----------- , 或
. . . . 2

. . 5
= -- (-1+i)
. . 2

. . (1+2i)(1+3i) . . . 5
∴ ----------------- = ---- (-1+i) - - - - - - - - - - - - - (#)
. . . .(1-i)(1+i) . . . . 2

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
5)
2 (1+2i)(1+3i)
------------------- - 5(2i³ + i⁶) + (7i⁹ + 2i¹²) = X(13+11i+12i²)
. . (1-i)(1+i)

. . . 5
(2) -- (-1+i) - 5(2i³ + i⁶) + (7i⁹ + 2i¹²) = X(13+11i+12i²) - - - - - - - - - - - - - 從 (#)
. . . 2

5(-1+i) - 5(2i³ + i⁶) + (7i⁹ + 2i¹²) = X(13+11i+12i²)
-5+5i-5[2(-i) + (-1)] + 7i + 2 = X [13+11i+12(-1)] - - - - - - - - - - - - - [請參考(註) -- 在答案下方]
-5+5i +10i + 5 + 7i + 2 = X (1+11i)
2+22i = X (1+11i)
2(1+11i) = X (1+11i)
∴ X = 2


~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 完 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

註 :
===
i = √-1
i² = √-1 √-1 = -1
i³ = i²× i = -1× i = -i
i⁴ = i²× i² = -1× -1 = 1
. . . . . . . . . . . . . . . . . .
i⁵ = i⁴ × i = 1× i = i
i⁶ = i⁴ × i² = 1× i² = i² = -1
i^7 = i⁴ × i³ = . . . . = -i
i^8 = i⁴ × i⁴ = . . . . = 1
. . . . . . . . . . . . . . . . . .
i 的次方 是一個循環 pattern :- i, -1, -i, 1

eg. i^13 = i⁴ × i⁴ × i⁴ × i = 1×1×1× i = i
eg. i^18 = i⁴ × i⁴ × i⁴ × i⁴ × i² = 1×1×1× 1 × -1 = -1

記著 :-
====
i² = -1
i⁴ = 1
很好用 ！！！