【数学I】例題1.2.5：対称式x^n+y^nの値（One More）★★★

% 例題I1.2.5：対称式$x^n+y^n$の値 （One More）★★★
$x=\frac{2}{\sqrt{3}+\sqrt{2}},y=\frac{2}{\sqrt{3}-\sqrt{2}}$のとき,次の値を求めよ. (1)$x+y$(2)$xy$(3)$x^2+y^2$(4)$x^3+y^3$(5)$x^4+y^4$(6)$x^5+y^5$

% 解答（例題I1.2.5）
(1)$\begin{aligned} x+y&=\dfrac{2}{\sqrt{3}+\sqrt{2}}+\dfrac{2}{\sqrt{3}-\sqrt{2}}=\dfrac{2(\sqrt{3}-\sqrt{2})+2(\sqrt{3}+\sqrt{2})}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}\\ &=\dfrac{4\sqrt{3}}{3-2}=4\sqrt{3} \end{aligned}$(2)$xy=\dfrac{2}{\sqrt{3}+\sqrt{2}} \cdot \dfrac{2}{\sqrt{3}-\sqrt{2}}=\frac{4}{3-2}=4$(3)$x^2+y^2=(x+y)^2-2xy=(4\sqrt{3})^2-2 \cdot 4=48-8=40$(4)$\begin{aligned} x^3+y^3&=(x+y)^3-3xy(x+y)=(4\sqrt{3})^3-3 \cdot 4 \cdot 4\sqrt{3}\\ &=192\sqrt{3}-48\sqrt{3}=144\sqrt{3} \end{aligned}$別解：$x^3+y^3=(x+y)(x^2-xy+y^2)=4\sqrt{3} \cdot (40-4)=144\sqrt{3}$(5)$x^4+y^4=(x^2+y^2)^2-2x^2y^2=40^2-2 \cdot 4^2=1600-32=1568$(6)$\begin{aligned} x^5+y^5&=(x^2+y^2)(x^3+y^3)-x^2y^3-x^3y^2\\ &=(x^2+y^2)(x^3+y^3)-(xy)^2(x+y)\\ &=40 \cdot 144\sqrt{3}-4^2 \cdot 4\sqrt{3}=5760\sqrt{3}-64\sqrt{3}=5696\sqrt{3} \end{aligned}$

% 問題I1.2.5
$x=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}},y=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$のとき,次の値を求めよ. (1)$x+y$(2)$xy$(3)$x^2+y^2$(4)$x^3+y^3$(5)$x^4+y^4$

% 解答I1.2.5
(1)$\begin{aligned} x+y&=\dfrac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}+\dfrac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} =\dfrac{(\sqrt{5}-\sqrt{3})^2+(\sqrt{5}+\sqrt{3})^2}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})}\\ &=\dfrac{(5-2\sqrt{15}+3)+(5+2\sqrt{15}+3)}{5-3} =8 \end{aligned}$(2)$xy=\dfrac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}} \cdot \dfrac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}=1$(3)$x^2+y^2=(x+y)^2-2xy=8^2-2 \cdot 1=64-2=62$(4)$x^3+y^3=(x+y)^3-3xy(x+y)=8^3-3 \cdot 1 \cdot 8=512-24=488$別解：$x^3+y^3=(x+y)(x^2-xy+y^2)=8 \cdot (62-1)=488$(5)$x^4+y^4=(x^2+y^2)^2-2x^2y^2=62^2-2 \cdot 1^2=3844-2=3842$