The expression `(1)/(sqrt(x+2sqrt(x-1)))+(1)/(sqrt(x-2sqrt(x-1)))` simplifies to:

Options

A. `(2)/(3-x) if 1 lt x lt 2`
B. `(2)/(2-x) if 1 lt x lt 2`
C. `(2 sqrt(x-1))/((x-2))if x gt 2`
D. `(2 sqrt(x-1))/((x+2))if x gt 2`

Answer

To simplify the expression \(\frac{1}{\sqrt{x + 2\sqrt{x - 1}}} + \frac{1}{\sqrt{x - 2\sqrt{x - 1}}}\), we will follow these steps:

### Step 1: Rewrite the square roots
We start by rewriting the terms under the square roots. We can express \(x + 2\sqrt{x - 1}\) and \(x - 2\sqrt{x - 1}\) in a more manageable form.

\[ \sqrt{x + 2\sqrt{x - 1}} = \sqrt{(\sqrt{x - 1} + 1)^2} = \sqrt{x - 1} + 1 \]
\[ \sqrt{x - 2\sqrt{x - 1}} = \sqrt{(\sqrt{x - 1} - 1)^2} = \sqrt{x - 1} - 1 \]

### Step 2: Substitute back into the expression
Now we substitute these back into the original expression:

\[ \frac{1}{\sqrt{x + 2\sqrt{x - 1}}} + \frac{1}{\sqrt{x - 2\sqrt{x - 1}}} = \frac{1}{\sqrt{x - 1} + 1} + \frac{1}{\sqrt{x - 1} - 1} \]

### Step 3: Find a common denominator
To combine these fractions, we need a common denominator:

\[ \text{Common denominator} = (\sqrt{x - 1} + 1)(\sqrt{x - 1} - 1) = (\sqrt{x - 1})^2 - 1^2 = (x - 1) - 1 = x - 2 \]

### Step 4: Combine the fractions
Now we can combine the fractions:

\[ \frac{(\sqrt{x - 1} - 1) + (\sqrt{x - 1} + 1)}{x - 2} = \frac{2\sqrt{x - 1}}{x - 2} \]

### Step 5: Final expression
Thus, the simplified expression is:

\[ \frac{2\sqrt{x - 1}}{x - 2} \]

### Summary
The expression \(\frac{1}{\sqrt{x + 2\sqrt{x - 1}}} + \frac{1}{\sqrt{x - 2\sqrt{x - 1}}}\) simplifies to \(\frac{2\sqrt{x - 1}}{x - 2}\).

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