Let's break down the given expression step by step:
The expression is:
[tex]\[ \sqrt{3} + 2 \sqrt{3 + \sqrt{7}} + \sqrt{7} \][/tex]
### Step 1: Calculate \( \sqrt{3} \)
The first part of the expression is \( \sqrt{3} \).
[tex]\[ \sqrt{3} \approx 1.732 \][/tex]
### Step 2: Calculate \( \sqrt{7} \)
Next, we calculate \( \sqrt{7} \).
[tex]\[ \sqrt{7} \approx 2.646 \][/tex]
### Step 3: Calculate the inner expression \( \sqrt{3 + \sqrt{7}} \)
Now, let's calculate the inner part.
[tex]\[ \sqrt{3 + \sqrt{7}} \][/tex]
We already know \( \sqrt{7} \approx 2.646 \), thus:
[tex]\[ 3 + \sqrt{7} \approx 3 + 2.646 = 5.646 \][/tex]
Then, we find the square root of this sum:
[tex]\[ \sqrt{5.646} \approx 2.376 \][/tex]
### Step 4: Multiply the inner result by 2
The expression inside the outer square root is multiplied by 2:
[tex]\[ 2 \sqrt{3 + \sqrt{7}} \approx 2 \times 2.376 = 4.752 \][/tex]
### Step 5: Add all parts together
Finally, we sum up all parts of the expression:
[tex]\[ \sqrt{3} + 2 \sqrt{3 + \sqrt{7}} + \sqrt{7} \approx 1.732 + 4.752 + 2.646 \][/tex]
Adding these values together gives:
[tex]\[ 1.732 + 4.752 + 2.646 = 9.130 \][/tex]
Therefore, the value of the expression:
[tex]\[ \sqrt{3} + 2 \sqrt{3 + \sqrt{7}} + \sqrt{7} \approx 9.130 \][/tex]
