If `x = [1//sqrt5 + sqrt3) ], y = [1// (sqrt7 + sqrt5)] and z = [1 // sqrt7 + sqrt3 ) ],` then what is the value of `(x + y + z)` ?

Options

A. `3//4 (sqrt7 - sqrt3)`
B. `sqrt7 - sqrt3`
C. `sqrt7 + sqrt5`
D. `1//2 (sqrt7 + sqrt3)`

Answer

To solve the problem, we need to find the value of \( x + y + z \) where:

\[ x = \frac{1}{\sqrt{5} + \sqrt{3}}, \quad y = \frac{1}{\sqrt{7} + \sqrt{5}}, \quad z = \frac{1}{\sqrt{7} + \sqrt{3}} \]

### Step 1: Simplify \( x \)

To simplify \( x \), we will rationalize the denominator:

\[ x = \frac{1}{\sqrt{5} + \sqrt{3}} \cdot \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}} = \frac{\sqrt{5} - \sqrt{3}}{(\sqrt{5})^2 - (\sqrt{3})^2} \]

Calculating the denominator:

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

Thus, we have:

\[ x = \frac{\sqrt{5} - \sqrt{3}}{2} \]

### Step 2: Simplify \( y \)

Next, we simplify \( y \):

\[ y = \frac{1}{\sqrt{7} + \sqrt{5}} \cdot \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} - \sqrt{5}} = \frac{\sqrt{7} - \sqrt{5}}{(\sqrt{7})^2 - (\sqrt{5})^2} \]

Calculating the denominator:

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

Thus, we have:

\[ y = \frac{\sqrt{7} - \sqrt{5}}{2} \]

### Step 3: Simplify \( z \)

Now, we simplify \( z \):

\[ z = \frac{1}{\sqrt{7} + \sqrt{3}} \cdot \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} - \sqrt{3}} = \frac{\sqrt{7} - \sqrt{3}}{(\sqrt{7})^2 - (\sqrt{3})^2} \]

Calculating the denominator:

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

Thus, we have:

\[ z = \frac{\sqrt{7} - \sqrt{3}}{4} \]

### Step 4: Combine \( x, y, z \)

Now, we can add \( x, y, z \):

\[ x + y + z = \frac{\sqrt{5} - \sqrt{3}}{2} + \frac{\sqrt{7} - \sqrt{5}}{2} + \frac{\sqrt{7} - \sqrt{3}}{4} \]

To add these fractions, we need a common denominator. The least common multiple of 2 and 4 is 4. We rewrite \( x \) and \( y \):

\[ x + y + z = \frac{2(\sqrt{5} - \sqrt{3})}{4} + \frac{2(\sqrt{7} - \sqrt{5})}{4} + \frac{\sqrt{7} - \sqrt{3}}{4} \]

Combining the numerators:

\[ = \frac{2(\sqrt{5} - \sqrt{3}) + 2(\sqrt{7} - \sqrt{5}) + (\sqrt{7} - \sqrt{3})}{4} \]

Distributing:

\[ = \frac{2\sqrt{5} - 2\sqrt{3} + 2\sqrt{7} - 2\sqrt{5} + \sqrt{7} - \sqrt{3}}{4} \]

Notice that \( 2\sqrt{5} \) cancels out:

\[ = \frac{3\sqrt{7} - 3\sqrt{3}}{4} \]

Factoring out the common term:

\[ = \frac{3(\sqrt{7} - \sqrt{3})}{4} \]

### Final Result

Thus, the value of \( x + y + z \) is:

\[ \frac{3(\sqrt{7} - \sqrt{3})}{4} \]