If $x + \frac{1}{x} = 2$, then find the value of

$$
x^{2723} + \frac{1}{x^{3356}}.
$$

Question

If $x + \frac{1}{x} = 2$, then find the value of

$$
x^{2723} + \frac{1}{x^{3356}}.
$$

Accepted Answer

We are given:

$$
x + \frac{1}{x} = 2
$$

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### Step 1: Recognize the identity

The equation

$$
x + \frac{1}{x} = 2
$$

holds **if and only if**

$$
x = 1
$$

because

$$
1 + \frac{1}{1} = 2
$$

(Other complex roots also satisfy the original equation, but in most competitive exams, $x$ is taken as a real positive number unless stated otherwise.)

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### Step 2: Use the value of $x$

If $x = 1$, then:

$$
x^{2723} = 1^{2723} = 1, \quad \frac{1}{x^{3356}} = \frac{1}{1^{3356}} = 1
$$

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### Final Answer:

$$
x^{2723} + \frac{1}{x^{3356}} = 1 + 1 = \boxed{2}
$$

Answer

Answer

Answer