Find the positive square root of the following :
`10+2sqrt(6)+sqrt(60)+2sqrt(10)`

Answer

To find the positive square root of the expression \(10 + 2\sqrt{6} + \sqrt{60} + 2\sqrt{10}\), we will simplify it step by step.

### Step 1: Rewrite \(\sqrt{60}\)
We can express \(\sqrt{60}\) in a simpler form:
\[ \sqrt{60} = \sqrt{4 \times 15} = 2\sqrt{15} \]
So, we can rewrite the expression as:
\[ 10 + 2\sqrt{6} + 2\sqrt{15} + 2\sqrt{10} \]

### Step 2: Group the terms
Now, we can group the terms for clarity:
\[ 10 + 2\sqrt{6} + 2\sqrt{15} + 2\sqrt{10} \]

### Step 3: Rewrite \(10\) in a useful form
We can express \(10\) as:
\[ 10 = 2^2 + 3 + 5 \]
Thus, we can rewrite the expression as:
\[ (2^2) + 3 + 5 + 2\sqrt{6} + 2\sqrt{15} + 2\sqrt{10} \]

### Step 4: Identify the perfect square
Notice that we can express the entire expression as a perfect square:
\[ ( \sqrt{2} + \sqrt{3} + \sqrt{5} )^2 \]
This expands to:
\[ (\sqrt{2})^2 + (\sqrt{3})^2 + (\sqrt{5})^2 + 2(\sqrt{2})(\sqrt{3}) + 2(\sqrt{3})(\sqrt{5}) + 2(\sqrt{5})(\sqrt{2}) \]
Which is:
\[ 2 + 3 + 5 + 2\sqrt{6} + 2\sqrt{15} + 2\sqrt{10} \]

### Step 5: Conclusion
Thus, we have:
\[ 10 + 2\sqrt{6} + 2\sqrt{15} + 2\sqrt{10} = (\sqrt{2} + \sqrt{3} + \sqrt{5})^2 \]
Taking the positive square root gives us:
\[ \sqrt{10 + 2\sqrt{6} + \sqrt{60} + 2\sqrt{10}} = \sqrt{2} + \sqrt{3} + \sqrt{5} \]

### Final Answer
The positive square root of the expression is:
\[ \sqrt{2} + \sqrt{3} + \sqrt{5} \]