次の複素数の計算をしなさい。 (1) $3i \times 4i$ (2) $i(4+i)$ (3) $(1+i)(3+2i)$ (4) $(2+i)(3-4i)$ (5) $(5+i)^2$ (6) $(4+3i)(4-3i)$

1. 問題の内容

次の複素数の計算をしなさい。

(1) $3i \times 4i$

(2) $i(4+i)$

(3) $(1+i)(3+2i)$

(4) $(2+i)(3-4i)$

(5) $(5+i)^2$

(6) $(4+3i)(4-3i)$

2. 解き方の手順

(1) $3i \times 4i = 12i^2 = 12(-1) = -12$

(2) $i(4+i) = 4i + i^2 = 4i - 1 = -1 + 4i$

(3) $(1+i)(3+2i) = 1(3) + 1(2i) + i(3) + i(2i) = 3 + 2i + 3i + 2i^2 = 3 + 5i + 2(-1) = 3 + 5i - 2 = 1 + 5i$

(4) $(2+i)(3-4i) = 2(3) + 2(-4i) + i(3) + i(-4i) = 6 - 8i + 3i - 4i^2 = 6 - 5i - 4(-1) = 6 - 5i + 4 = 10 - 5i$

(5) $(5+i)^2 = (5+i)(5+i) = 5(5) + 5(i) + i(5) + i(i) = 25 + 5i + 5i + i^2 = 25 + 10i + (-1) = 24 + 10i$

(6) $(4+3i)(4-3i) = 4(4) + 4(-3i) + 3i(4) + 3i(-3i) = 16 - 12i + 12i - 9i^2 = 16 - 9(-1) = 16 + 9 = 25$

3. 最終的な答え

(1) $-12$

(2) $-1 + 4i$

(3) $1 + 5i$

(4) $10 - 5i$

(5) $24 + 10i$

(6) $25$