If `x+(1)/(x)=2`, then find the value of `(sqrt(x)+(1)/(sqrt(x)))`

Options

A. `sqrt(2)`
B. 2
C. `sqrt(3)`
D. 1

Answer

To solve the equation \( x + \frac{1}{x} = 2 \) and find the value of \( \sqrt{x} + \frac{1}{\sqrt{x}} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ x + \frac{1}{x} = 2 \]

### Step 2: Square both sides
To find \( \sqrt{x} + \frac{1}{\sqrt{x}} \), we can square \( \sqrt{x} + \frac{1}{\sqrt{x}} \):
\[ \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^2 = x + 2 + \frac{1}{x} \]

### Step 3: Substitute the value from the given equation
From the given equation, we know that \( x + \frac{1}{x} = 2 \). Therefore, we can substitute this into our squared equation:
\[ \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^2 = 2 + 2 \]

### Step 4: Simplify the equation
Now, simplify the right-hand side:
\[ \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^2 = 4 \]

### Step 5: Take the square root of both sides
Taking the square root of both sides gives us:
\[ \sqrt{x} + \frac{1}{\sqrt{x}} = 2 \quad \text{or} \quad \sqrt{x} + \frac{1}{\sqrt{x}} = -2 \]

### Step 6: Determine the valid solution
Since \( \sqrt{x} \) and \( \frac{1}{\sqrt{x}} \) are both positive for real \( x \), we discard the negative solution. Thus, we have:
\[ \sqrt{x} + \frac{1}{\sqrt{x}} = 2 \]

### Final Answer
The value of \( \sqrt{x} + \frac{1}{\sqrt{x}} \) is:
\[ \boxed{2} \]