Find the square root of
`x+y+z+2sqrt(xz+yz)`.

Answer

To find the square root of the expression \( x + y + z + 2\sqrt{xz + yz} \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ x + y + z + 2\sqrt{xz + yz} \]

### Step 2: Identify Perfect Squares
Notice that \( \sqrt{xz + yz} \) can be factored. We can rewrite \( xz + yz \) as \( z(x + y) \):
\[ x + y + z + 2\sqrt{z(x + y)} \]

### Step 3: Recognize the Form of a Perfect Square
The expression \( x + y + z + 2\sqrt{z(x + y)} \) resembles the expansion of a binomial square. Specifically, it can be expressed as:
\[ (\sqrt{x + y} + \sqrt{z})^2 \]
This is because:
\[ (\sqrt{x + y} + \sqrt{z})^2 = (\sqrt{x + y})^2 + (\sqrt{z})^2 + 2\sqrt{z}\sqrt{x + y} = (x + y) + z + 2\sqrt{z(x + y)} \]

### Step 4: Take the Square Root
Now that we have expressed the original expression as a square, we can take the square root:
\[ \sqrt{x + y + z + 2\sqrt{xz + yz}} = \sqrt{(\sqrt{x + y} + \sqrt{z})^2} \]
This simplifies to:
\[ \sqrt{x + y} + \sqrt{z} \]

### Final Answer
Thus, the square root of the expression is:
\[ \sqrt{x + y} + \sqrt{z} \]

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