Questions: When x + y = 4, what is x^4 + y^4? Given that xy = 2.

When x + y = 4, what is x^4 + y^4? Given that xy = 2.
Transcript text: $\left\{\begin{array}{l}x+y=4 \text { 일때, } x^{4}+y^{4}=? \\ x y=2\end{array}\right.$
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Solution

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Step 1: 주어진 값

주어진 연립 방정식은 다음과 같다:
\[ \left\{\begin{array}{l}x+y=4 \\ xy=2\end{array}\right. \]

Step 2: \(x^2 + y^2\) 계산

\(x^2 + y^2\)는 다음과 같이 계산할 수 있다:
\[ x^2 + y^2 = (x+y)^2 - 2xy = 4^2 - 2 \cdot 2 = 16 - 4 = 12 \]

Step 3: \(x^4 + y^4\) 계산

\(x^4 + y^4\)는 다음과 같이 계산할 수 있다:
\[ x^4 + y^4 = (x^2 + y^2)^2 - 2(xy)^2 = 12^2 - 2 \cdot 2^2 = 144 - 8 = 136 \]

최종 답변

\(\boxed{136}\)

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