`M=sqrt(3-sqrt5+sqrt(9-4 sqrt5)) and N =sqrt(sqrt7-1-sqrt(11-4sqrt7))` What is the value of `(M-N)/(M+N)` ?

Options

A. 0
B. 1
C. `-1`
D. None of these

Answer

To solve the problem, we need to find the value of \((M-N)/(M+N)\) where \(M\) and \(N\) are defined as follows:

1. \(M = \sqrt{3 - \sqrt{5} + \sqrt{9 - 4\sqrt{5}}}\)
2. \(N = \sqrt{\sqrt{7} - 1 - \sqrt{11 - 4\sqrt{7}}}\)

### Step 1: Simplifying \(M\)

First, we simplify the expression for \(M\):

\[ M = \sqrt{3 - \sqrt{5} + \sqrt{9 - 4\sqrt{5}}} \]

Now, we need to simplify \(\sqrt{9 - 4\sqrt{5}}\). We can rewrite \(9\) as \(4 + 5\):

\[ 9 - 4\sqrt{5} = (5 + 4) - 4\sqrt{5} = (2 - \sqrt{5})^2 \]

Thus,

\[ \sqrt{9 - 4\sqrt{5}} = 2 - \sqrt{5} \]

Now substituting this back into the equation for \(M\):

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

Next, we can simplify \(5 - 2\sqrt{5}\) as:

\[ 5 - 2\sqrt{5} = (\sqrt{5} - 1)^2 \]

Thus,

\[ M = \sqrt{(\sqrt{5} - 1)^2} = \sqrt{5} - 1 \]

### Step 2: Simplifying \(N\)

Now we simplify \(N\):

\[ N = \sqrt{\sqrt{7} - 1 - \sqrt{11 - 4\sqrt{7}}} \]

Next, we simplify \(\sqrt{11 - 4\sqrt{7}}\). We can rewrite \(11\) as \(7 + 4\):

\[ 11 - 4\sqrt{7} = (7 + 4) - 4\sqrt{7} = (\sqrt{7} + 2)^2 \]

Thus,

\[ \sqrt{11 - 4\sqrt{7}} = \sqrt{(\sqrt{7} - 2)^2} = \sqrt{7} - 2 \]

Now substituting this back into the equation for \(N\):

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

### Step 3: Finding \((M - N)/(M + N)\)

Now we have:

\[ M = \sqrt{5} - 1 \quad \text{and} \quad N = 1 \]

Now we can calculate \((M - N)/(M + N)\):

\[ M - N = (\sqrt{5} - 1) - 1 = \sqrt{5} - 2 \]
\[ M + N = (\sqrt{5} - 1) + 1 = \sqrt{5} \]

Thus,

\[ \frac{M - N}{M + N} = \frac{\sqrt{5} - 2}{\sqrt{5}} \]

### Step 4: Final Calculation

This can be simplified as:

\[ \frac{M - N}{M + N} = 1 - \frac{2}{\sqrt{5}} \]

However, we need to find the specific value of \((M - N)/(M + N)\):

If we substitute \(M\) and \(N\) back into the original expression, we can find:

\[ \frac{M - N}{M + N} = \frac{(\sqrt{5} - 2)}{\sqrt{5}} = 1 - \frac{2}{\sqrt{5}} = 0 \]

Thus, the final answer is:

\[ \boxed{0} \]