If `sqrt(9+sqrt48-sqrt32-sqrt24)=sqrta-sqrtb+2`,where `a,b inN`, then find the value of a + b.

Answer

To solve the equation \( \sqrt{9 + \sqrt{48} - \sqrt{32} - \sqrt{24}} = \sqrt{a} - \sqrt{b} + 2 \), where \( a, b \in \mathbb{N} \), we will simplify the left-hand side step by step.

### Step 1: Simplify the expression inside the square root
We start with the left-hand side (LHS):
\[ \sqrt{9 + \sqrt{48} - \sqrt{32} - \sqrt{24}} \]

### Step 2: Simplify the square roots
First, we simplify the square roots:
- \( \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3} \)
- \( \sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2} \)
- \( \sqrt{24} = \sqrt{16 \cdot 1.5} = 4\sqrt{1.5} = 2\sqrt{6} \)

Now substituting these back into the expression:
\[ \sqrt{9 + 4\sqrt{3} - 4\sqrt{2} - 2\sqrt{6}} \]

### Step 3: Rewrite 9 as a sum of squares
Notice that \( 9 \) can be rewritten as:
\[ 9 = 3^2 + 0^2 + 0^2 \]

### Step 4: Group the terms
We can express the LHS as:
\[ \sqrt{(3 + 2 - \sqrt{2})^2} \]
where we have:
- \( a = 3 \)
- \( b = 2 \)

### Step 5: Compare with the right-hand side
The right-hand side (RHS) is given as:
\[ \sqrt{a} - \sqrt{b} + 2 \]
From our simplification, we can see that:
\[ \sqrt{3} - \sqrt{2} + 2 \]

### Step 6: Identify \( a \) and \( b \)
From the comparison:
- We have identified \( a = 3 \)
- We have identified \( b = 2 \)

### Step 7: Calculate \( a + b \)
Now, we need to find \( a + b \):
\[ a + b = 3 + 2 = 5 \]

Thus, the final answer is:
\[ \boxed{5} \]