The value of `sqrt((3sqrt(9)-3sqrt(8))(9+2sqrt(18)))` is :

Options

A. 3
B. 4
C. 2
D. 9

Answer

To solve the expression \( \sqrt{(3\sqrt{9} - 3\sqrt{8})(9 + 2\sqrt{18})} \), we will break it down step by step.

### Step 1: Simplify \( 3\sqrt{9} \) and \( 3\sqrt{8} \)
First, we calculate \( 3\sqrt{9} \) and \( 3\sqrt{8} \):
- \( \sqrt{9} = 3 \) so \( 3\sqrt{9} = 3 \times 3 = 9 \)
- \( \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \) so \( 3\sqrt{8} = 3 \times 2\sqrt{2} = 6\sqrt{2} \)

Thus, we have:
\[ 3\sqrt{9} - 3\sqrt{8} = 9 - 6\sqrt{2} \]

### Step 2: Simplify \( 9 + 2\sqrt{18} \)
Next, we simplify \( 2\sqrt{18} \):
- \( \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \) so \( 2\sqrt{18} = 2 \times 3\sqrt{2} = 6\sqrt{2} \)

Thus, we have:
\[ 9 + 2\sqrt{18} = 9 + 6\sqrt{2} \]

### Step 3: Substitute back into the expression
Now substituting back into the original expression:
\[ \sqrt{(9 - 6\sqrt{2})(9 + 6\sqrt{2})} \]

### Step 4: Apply the difference of squares formula
Using the difference of squares formula \( (a - b)(a + b) = a^2 - b^2 \):
- Here, \( a = 9 \) and \( b = 6\sqrt{2} \)

Calculating:
\[ (9 - 6\sqrt{2})(9 + 6\sqrt{2}) = 9^2 - (6\sqrt{2})^2 \]
\[ = 81 - (36 \times 2) = 81 - 72 = 9 \]

### Step 5: Take the square root
Now we take the square root of the result:
\[ \sqrt{9} = 3 \]

### Final Answer
Thus, the value of \( \sqrt{(3\sqrt{9} - 3\sqrt{8})(9 + 2\sqrt{18})} \) is \( 3 \).

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