`(6^(2)+7^(2)+8^(2)+9^(2)+10^(2))/(sqrt(7+4sqrt3)-sqrt(4+2sqrt3))=?`

Options

A. 330
B. 305
C. 355
D. 366

Answer

To solve the given problem step-by-step, we need to evaluate the expression:

\[ \frac{6^2 + 7^2 + 8^2 + 9^2 + 10^2}{\sqrt{7 + 4\sqrt{3}} - \sqrt{4 + 2\sqrt{3}}} \]

### Step 1: Calculate the numerator

First, we need to calculate the squares of the numbers in the numerator:

\[ 6^2 = 36, \quad 7^2 = 49, \quad 8^2 = 64, \quad 9^2 = 81, \quad 10^2 = 100 \]

Now, we sum these values:

\[ 36 + 49 + 64 + 81 + 100 = 330 \]

### Step 2: Simplify the denominator

Next, we simplify the denominator:

\[ \sqrt{7 + 4\sqrt{3}} - \sqrt{4 + 2\sqrt{3}} \]

#### Step 2.1: Simplify \(\sqrt{7 + 4\sqrt{3}}\)

We can express \(7 + 4\sqrt{3}\) as a perfect square. We find \(a\) and \(b\) such that:

\[ (a + b)^2 = a^2 + b^2 + 2ab \]

Let \(a = 2\) and \(b = \sqrt{3}\):

\[ (2 + \sqrt{3})^2 = 2^2 + (\sqrt{3})^2 + 2 \cdot 2 \cdot \sqrt{3} = 4 + 3 + 4\sqrt{3} = 7 + 4\sqrt{3} \]

Thus, we have:

\[ \sqrt{7 + 4\sqrt{3}} = 2 + \sqrt{3} \]

#### Step 2.2: Simplify \(\sqrt{4 + 2\sqrt{3}}\)

Similarly, we can express \(4 + 2\sqrt{3}\) as a perfect square. Let \(c = 1\) and \(d = \sqrt{3}\):

\[ (1 + \sqrt{3})^2 = 1^2 + (\sqrt{3})^2 + 2 \cdot 1 \cdot \sqrt{3} = 1 + 3 + 2\sqrt{3} = 4 + 2\sqrt{3} \]

Thus, we have:

\[ \sqrt{4 + 2\sqrt{3}} = 1 + \sqrt{3} \]

### Step 3: Substitute back into the denominator

Now we substitute these values back into the denominator:

\[ \sqrt{7 + 4\sqrt{3}} - \sqrt{4 + 2\sqrt{3}} = (2 + \sqrt{3}) - (1 + \sqrt{3}) = 2 + \sqrt{3} - 1 - \sqrt{3} = 1 \]

### Step 4: Final calculation

Now we can substitute the values back into the main expression:

\[ \frac{330}{1} = 330 \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{330} \]