Certainly! Let's solve the given problem step-by-step:
Given the equation [tex]\( x + \frac{1}{x} = 2 \)[/tex].
We want to find the value of [tex]\( \sqrt{x} + \sqrt{\frac{1}{x}} \)[/tex].
Let's define [tex]\( y = \sqrt{x} + \sqrt{\frac{1}{x}} \)[/tex].
Now, we'll square both sides of this equation:
[tex]\[ y^2 = \left(\sqrt{x} + \sqrt{\frac{1}{x}}\right)^2 \][/tex]
By expanding the right side, we get:
[tex]\[ y^2 = (\sqrt{x})^2 + 2 \cdot \sqrt{x} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{x}}\right)^2 \][/tex]
Simplifying each term, we have:
[tex]\[ y^2 = x + 2 \cdot \sqrt{x \cdot \frac{1}{x}} + \frac{1}{x} \][/tex]
Given that [tex]\( \sqrt{x \cdot \frac{1}{x}} = \sqrt{1} = 1 \)[/tex], we can further simplify:
[tex]\[ y^2 = x + \frac{1}{x} + 2 \][/tex]
Given in the problem, we know that:
[tex]\[ x + \frac{1}{x} = 2 \][/tex]
Therefore, substituting this back into our equation, we get:
[tex]\[ y^2 = 2 + 2 \][/tex]
Thus:
[tex]\[ y^2 = 4 \][/tex]
To solve for [tex]\( y \)[/tex], we take the square root of both sides:
[tex]\[ y = \sqrt{4} \][/tex]
Hence:
[tex]\[ y = 2 \][/tex]
Therefore, the value of [tex]\( \sqrt{x} + \sqrt{\frac{1}{x}} \)[/tex] is [tex]\( \boxed{2} \)[/tex].
