The value of `sqrt(2+sqrt3)+sqrt(2-sqrt(3)` is

Options

A. `2sqrt2`
B. `2sqrt3`
C. `sqrt6`
D. 6

Answer

To find the value of \( \sqrt{2 + \sqrt{3}} + \sqrt{2 - \sqrt{3}} \), we can simplify this expression step by step.

### Step 1: Simplify Each Square Root
Let's denote:
- \( a = \sqrt{2 + \sqrt{3}} \)
- \( b = \sqrt{2 - \sqrt{3}} \)

We want to find \( a + b \).

### Step 2: Calculate \( a^2 + b^2 \)
First, we will calculate \( a^2 + b^2 \):
\[ a^2 = 2 + \sqrt{3} \]
\[ b^2 = 2 - \sqrt{3} \]
Now, adding these two:
\[ a^2 + b^2 = (2 + \sqrt{3}) + (2 - \sqrt{3}) = 4 \]

### Step 3: Calculate \( ab \)
Next, we calculate \( ab \):
\[ ab = \sqrt{(2 + \sqrt{3})(2 - \sqrt{3})} \]
Using the difference of squares:
\[ ab = \sqrt{2^2 - (\sqrt{3})^2} = \sqrt{4 - 3} = \sqrt{1} = 1 \]

### Step 4: Use the Identity for \( (a + b)^2 \)
We can use the identity:
\[ (a + b)^2 = a^2 + b^2 + 2ab \]
Substituting the values we found:
\[ (a + b)^2 = 4 + 2 \cdot 1 = 4 + 2 = 6 \]

### Step 5: Take the Square Root
Now, we take the square root of both sides to find \( a + b \):
\[ a + b = \sqrt{6} \]

Thus, the value of \( \sqrt{2 + \sqrt{3}} + \sqrt{2 - \sqrt{3}} \) is \( \sqrt{6} \).

### Final Answer
\[ \sqrt{2 + \sqrt{3}} + \sqrt{2 - \sqrt{3}} = \sqrt{6} \]

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