`((sqrt(2)-1)/(sqrt(2) +1) + (sqrt(2) +1)/(sqrt(2)-1))/((sqrt(3)-1)/(sqrt(3)+1) + (sqrt(3) +1)/(sqrt(3)-1))` equals:

Options

A. `2/3`
B. 4
C. `3/2`
D. 6

Answer

To solve the expression

\[ \frac{\left(\frac{\sqrt{2}-1}{\sqrt{2}+1} + \frac{\sqrt{2}+1}{\sqrt{2}-1}\right)}{\left(\frac{\sqrt{3}-1}{\sqrt{3}+1} + \frac{\sqrt{3}+1}{\sqrt{3}-1}\right)}, \]

we will simplify both the numerator and the denominator step by step.

### Step 1: Simplify the Numerator

The numerator is

\[ \frac{\sqrt{2}-1}{\sqrt{2}+1} + \frac{\sqrt{2}+1}{\sqrt{2}-1}. \]

To combine these fractions, we will find a common denominator. The common denominator is \((\sqrt{2}+1)(\sqrt{2}-1)\).

Now, we can rewrite the expression:

\[ \frac{(\sqrt{2}-1)^2 + (\sqrt{2}+1)^2}{(\sqrt{2}+1)(\sqrt{2}-1)}. \]

### Step 2: Expand the Numerator

Now, we will expand \((\sqrt{2}-1)^2\) and \((\sqrt{2}+1)^2\):

\[ (\sqrt{2}-1)^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2}, \]
\[ (\sqrt{2}+1)^2 = 2 + 2\sqrt{2} + 1 = 3 + 2\sqrt{2}. \]

Adding these two results gives:

\[ (3 - 2\sqrt{2}) + (3 + 2\sqrt{2}) = 6. \]

### Step 3: Simplify the Denominator

Now, we will simplify the denominator:

\[ \frac{\sqrt{3}-1}{\sqrt{3}+1} + \frac{\sqrt{3}+1}{\sqrt{3}-1}. \]

Similarly, we find a common denominator, which is \((\sqrt{3}+1)(\sqrt{3}-1)\):

\[ \frac{(\sqrt{3}-1)^2 + (\sqrt{3}+1)^2}{(\sqrt{3}+1)(\sqrt{3}-1)}. \]

### Step 4: Expand the Denominator

Now, we will expand \((\sqrt{3}-1)^2\) and \((\sqrt{3}+1)^2\):

\[ (\sqrt{3}-1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3}, \]
\[ (\sqrt{3}+1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3}. \]

Adding these two results gives:

\[ (4 - 2\sqrt{3}) + (4 + 2\sqrt{3}) = 8. \]

### Step 5: Combine the Results

Now we can combine the results from the numerator and denominator:

\[ \frac{6}{(\sqrt{2}+1)(\sqrt{2}-1)} \div \frac{8}{(\sqrt{3}+1)(\sqrt{3}-1)}. \]

This simplifies to:

\[ \frac{6}{8} \cdot \frac{(\sqrt{3}+1)(\sqrt{3}-1)}{(\sqrt{2}+1)(\sqrt{2}-1)} = \frac{3}{4} \cdot \frac{(\sqrt{3}+1)(\sqrt{3}-1)}{(\sqrt{2}+1)(\sqrt{2}-1)}. \]

### Step 6: Calculate the Final Value

Now, we know that:

\[ (\sqrt{3}+1)(\sqrt{3}-1) = 3 - 1 = 2, \]
\[ (\sqrt{2}+1)(\sqrt{2}-1) = 2 - 1 = 1. \]

Thus, we have:

\[ \frac{3}{4} \cdot \frac{2}{1} = \frac{3 \cdot 2}{4} = \frac{6}{4} = \frac{3}{2}. \]

### Final Answer

The final answer is

\[ \frac{3}{2}. \]