Find the square root of : `21+3sqrt(8)-6sqrt(3)-6sqrt(7)-sqrt(24)-sqrt(56)+2sqrt(21)`.

Answer

To find the square root of the expression \( 21 + 3\sqrt{8} - 6\sqrt{3} - 6\sqrt{7} - \sqrt{24} - \sqrt{56} + 2\sqrt{21} \), we will simplify it step by step.

### Step 1: Simplify the square root terms
First, we simplify the square root terms in the expression:

1. \( \sqrt{8} = 2\sqrt{2} \)
2. \( \sqrt{24} = 2\sqrt{6} \)
3. \( \sqrt{56} = 2\sqrt{14} \)

Substituting these back into the expression gives us:

\[ 21 + 3(2\sqrt{2}) - 6\sqrt{3} - 6\sqrt{7} - 2\sqrt{6} - 2\sqrt{14} + 2\sqrt{21} \]

This simplifies to:

\[ 21 + 6\sqrt{2} - 6\sqrt{3} - 6\sqrt{7} - 2\sqrt{6} - 2\sqrt{14} + 2\sqrt{21} \]

### Step 2: Combine like terms
Now, we can combine the constant and square root terms:

The constant term is \( 21 \).

The square root terms are:

\[ 6\sqrt{2} - 6\sqrt{3} - 6\sqrt{7} - 2\sqrt{6} - 2\sqrt{14} + 2\sqrt{21} \]

### Step 3: Group terms for easier handling
We can group the square root terms:

\[ = 6(\sqrt{2} - \sqrt{3} - \sqrt{7}) - 2(\sqrt{6} + \sqrt{14}) + 2\sqrt{21} \]

### Step 4: Rewrite the expression
Now, let's rewrite the entire expression:

\[ = 21 + 6(\sqrt{2} - \sqrt{3} - \sqrt{7}) - 2(\sqrt{6} + \sqrt{14}) + 2\sqrt{21} \]

### Step 5: Assume the form of the square root
Assume that the square root of the entire expression can be written in the form:

\[ \sqrt{(a + b + c)^2} \]

Where \( a, b, c \) are terms we need to find.

### Step 6: Set up the equation
We can set:

\[ M = \sqrt{2} - \sqrt{3} - \sqrt{7} \]

Then we can express the entire expression as:

\[ (3 + M)^2 \]

### Step 7: Expand the square
Expanding \( (3 + M)^2 \):

\[ = 9 + 6M + M^2 \]

### Step 8: Substitute back for M
Substituting back \( M = \sqrt{2} - \sqrt{3} - \sqrt{7} \):

\[ = 9 + 6(\sqrt{2} - \sqrt{3} - \sqrt{7}) + (\sqrt{2} - \sqrt{3} - \sqrt{7})^2 \]

### Step 9: Final simplification
The final expression simplifies to:

\[ = 9 + 6(\sqrt{2} - \sqrt{3} - \sqrt{7}) + (2 - 3 - 7 + 2\sqrt{6} + 2\sqrt{14} + 2\sqrt{21}) \]

### Step 10: Calculate the square root
Finally, we can conclude that the square root of the original expression is:

\[ \sqrt{(3 + \sqrt{2} - \sqrt{3} - \sqrt{7})^2} = 3 + \sqrt{2} - \sqrt{3} - \sqrt{7} \]

### Final Answer:
\[ \sqrt{21 + 3\sqrt{8} - 6\sqrt{3} - 6\sqrt{7} - \sqrt{24} - \sqrt{56} + 2\sqrt{21}} = 3 + \sqrt{2} - \sqrt{3} - \sqrt{7} \]