If x + 1/x = sqrt(2) find the value of x ^ 80 + x ^ 76 + x ^ 72 + x ^ 65 + x ^ 64 + 4

We are given the equation \( x + \frac{1}{x} = \sqrt{2} \), and we need to find the value of \( x^{80} + x^{76} + x^{72} + x^{65} + x^{64} + 4 \).

### Step 1: Analyze the given equation
We start with the equation:
\[
x + \frac{1}{x} = \sqrt{2}
\]
Let's square both sides to get a simpler form involving powers of \( x \):
\[
\left( x + \frac{1}{x} \right)^2 = (\sqrt{2})^2
\]
\[
x^2 + 2 + \frac{1}{x^2} = 2
\]
Simplifying, we get:
\[
x^2 + \frac{1}{x^2} = 0
\]
Thus:
\[
x^2 = -\frac{1}{x^2}
\]
or equivalently:
\[
x^4

If x + 1/x = sqrt(2), find the value of x ^ 80 + x ^ 76 + x ^ 72 + x ^ 65 + x ^ 64 + 4

Explore how to solve a complex equation involving powers of variables, given x + 1/x = sqrt(2). This detailed solution walks through key algebraic steps and simplifications, suitable for high school students in Grades 10-12.

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