To find the value of $x^{99} + \frac{1}{x^{99}}$ given the equation $x + \frac{1}{x} = 2$, we first need to determine the value of $x$. Let's follow these steps:

Solving the Initial Equation for x

We are given the equation:

To solve for $x$, we can eliminate the fraction by multiplying the entire equation by $x$ (assuming $x \neq 0$):

Now, rearrange the terms to form a standard quadratic equation:

This is a perfect square trinomial, which can be factored as:

Taking the square root of both sides gives:

Therefore, the value of $x$ is:

Calculating x⁹⁹ + 1/x⁹⁹

Now that we have found $x = 1$, we can substitute this value into the expression we need to evaluate:

Substitute $x = 1$ into the expression:

Calculate the powers:

• $1^{99} = 1$
• $\frac{1}{1^{99}} = \frac{1}{1} = 1$

Now, add the results:

So, the value of $x^{99} + \frac{1}{x^{99}}$ is 2.

Final Answer Verification

The steps clearly show that when $x + \frac{1}{x} = 2$, the value of $x$ must be 1. Substituting $x=1$ into $x^{99} + \frac{1}{x^{99}}$ yields $1^{99} + \frac{1}{1^{99}} = 1 + 1 = 2$. This confirms our result.