The value of `sqrt(11+2sqrt(30))-1/(sqrt(11+2sqrt(30)))` is

Options

A. `2sqrt5`
B. `2sqrt6`
C. `1+sqrt6`
D. `1+sqrt5`

Answer

To solve the expression \( \sqrt{11 + 2\sqrt{30}} - \frac{1}{\sqrt{11 + 2\sqrt{30}}} \), we will follow these steps:

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

We start with the expression inside the square root:
\[ \sqrt{11 + 2\sqrt{30}} \]
We can express \( 11 + 2\sqrt{30} \) in the form \( (\sqrt{a} + \sqrt{b})^2 \).

To find \( a \) and \( b \), we set:
\[ a + b = 11 \quad \text{and} \quad 2\sqrt{ab} = 2\sqrt{30} \]
From \( 2\sqrt{ab} = 2\sqrt{30} \), we have:
\[ \sqrt{ab} = \sqrt{30} \implies ab = 30 \]

Now, we need to solve the system:
1. \( a + b = 11 \)
2. \( ab = 30 \)

### Step 2: Solve for \( a \) and \( b \)

Let \( a \) and \( b \) be the roots of the quadratic equation:
\[ x^2 - (a+b)x + ab = 0 \implies x^2 - 11x + 30 = 0 \]
Using the quadratic formula:
\[ x = \frac{11 \pm \sqrt{11^2 - 4 \cdot 30}}{2} = \frac{11 \pm \sqrt{121 - 120}}{2} = \frac{11 \pm 1}{2} \]
This gives us:
\[ x = \frac{12}{2} = 6 \quad \text{and} \quad x = \frac{10}{2} = 5 \]
Thus, \( a = 6 \) and \( b = 5 \).

### Step 3: Substitute back into the square root

Now we can rewrite:
\[ \sqrt{11 + 2\sqrt{30}} = \sqrt{(\sqrt{6} + \sqrt{5})^2} = \sqrt{6} + \sqrt{5} \]

### Step 4: Substitute into the original expression

Now substituting back into the original expression:
\[ \sqrt{11 + 2\sqrt{30}} - \frac{1}{\sqrt{11 + 2\sqrt{30}}} = (\sqrt{6} + \sqrt{5}) - \frac{1}{\sqrt{6} + \sqrt{5}} \]

### Step 5: Simplify the fraction

To simplify \( \frac{1}{\sqrt{6} + \sqrt{5}} \), we multiply the numerator and denominator by the conjugate:
\[ \frac{1}{\sqrt{6} + \sqrt{5}} \cdot \frac{\sqrt{6} - \sqrt{5}}{\sqrt{6} - \sqrt{5}} = \frac{\sqrt{6} - \sqrt{5}}{(\sqrt{6})^2 - (\sqrt{5})^2} = \frac{\sqrt{6} - \sqrt{5}}{6 - 5} = \sqrt{6} - \sqrt{5} \]

### Step 6: Combine the terms

Now we combine:
\[ (\sqrt{6} + \sqrt{5}) - (\sqrt{6} - \sqrt{5}) = \sqrt{6} + \sqrt{5} - \sqrt{6} + \sqrt{5} = 2\sqrt{5} \]

### Final Answer

Thus, the value of the expression is:
\[ \boxed{2\sqrt{5}} \]