Value of `1/sqrt(11-2sqrt30)-3/sqrt(7-2sqrt10)-4/(sqrt(8+4sqrt3))` is

Options

A. 0
B. 1
C. 2
D. 4

Answer

To solve the expression \( \frac{1}{\sqrt{11 - 2\sqrt{30}}} - \frac{3}{\sqrt{7 - 2\sqrt{10}}} - \frac{4}{\sqrt{8 + 4\sqrt{3}}} \), we will simplify each term step by step.

### Step 1: Simplify \( \sqrt{11 - 2\sqrt{30}} \)

We can express \( 11 \) as \( \sqrt{6}^2 + \sqrt{5}^2 \) and \( 2\sqrt{30} \) as \( 2\sqrt{6 \cdot 5} \). Thus, we can rewrite:

\[ \sqrt{11 - 2\sqrt{30}} = \sqrt{(\sqrt{6} - \sqrt{5})^2} \]

Taking the square root gives us:

\[ \sqrt{11 - 2\sqrt{30}} = \sqrt{6} - \sqrt{5} \]

### Step 2: Simplify \( \sqrt{7 - 2\sqrt{10}} \)

We can express \( 7 \) as \( \sqrt{5}^2 + \sqrt{2}^2 \) and \( 2\sqrt{10} \) as \( 2\sqrt{5 \cdot 2} \). Thus, we can rewrite:

\[ \sqrt{7 - 2\sqrt{10}} = \sqrt{(\sqrt{5} - \sqrt{2})^2} \]

Taking the square root gives us:

\[ \sqrt{7 - 2\sqrt{10}} = \sqrt{5} - \sqrt{2} \]

### Step 3: Simplify \( \sqrt{8 + 4\sqrt{3}} \)

We can express \( 8 \) as \( \sqrt{6}^2 + \sqrt{2}^2 \) and \( 4\sqrt{3} \) as \( 2 \cdot 2\sqrt{3} \). Thus, we can rewrite:

\[ \sqrt{8 + 4\sqrt{3}} = \sqrt{(\sqrt{6} + \sqrt{2})^2} \]

Taking the square root gives us:

\[ \sqrt{8 + 4\sqrt{3}} = \sqrt{6} + \sqrt{2} \]

### Step 4: Substitute back into the original expression

Now substituting back into the original expression:

\[ \frac{1}{\sqrt{6} - \sqrt{5}} - \frac{3}{\sqrt{5} - \sqrt{2}} - \frac{4}{\sqrt{6} + \sqrt{2}} \]

### Step 5: Rationalize each term

1. **First term**: Multiply numerator and denominator by \( \sqrt{6} + \sqrt{5} \):

\[ \frac{1(\sqrt{6} + \sqrt{5})}{(\sqrt{6} - \sqrt{5})(\sqrt{6} + \sqrt{5})} = \frac{\sqrt{6} + \sqrt{5}}{1} = \sqrt{6} + \sqrt{5} \]

2. **Second term**: Multiply numerator and denominator by \( \sqrt{5} + \sqrt{2} \):

\[ \frac{-3(\sqrt{5} + \sqrt{2})}{(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})} = \frac{-3(\sqrt{5} + \sqrt{2})}{3} = -(\sqrt{5} + \sqrt{2}) \]

3. **Third term**: Multiply numerator and denominator by \( \sqrt{6} - \sqrt{2} \):

\[ \frac{-4(\sqrt{6} - \sqrt{2})}{(\sqrt{6} + \sqrt{2})(\sqrt{6} - \sqrt{2})} = \frac{-4(\sqrt{6} - \sqrt{2})}{4} = -(\sqrt{6} - \sqrt{2}) \]

### Step 6: Combine all terms

Now we combine all the simplified terms:

\[ (\sqrt{6} + \sqrt{5}) - (\sqrt{5} + \sqrt{2}) - (\sqrt{6} - \sqrt{2}) \]

This simplifies to:

\[ \sqrt{6} + \sqrt{5} - \sqrt{5} - \sqrt{2} - \sqrt{6} + \sqrt{2} \]

### Step 7: Final simplification

All terms cancel out:

\[ \sqrt{6} - \sqrt{6} + \sqrt{5} - \sqrt{5} + \sqrt{2} - \sqrt{2} = 0 \]

Thus, the value of the expression is:

\[ \boxed{0} \]