If `z=6-2sqrt(3)`, find the value of `(sqrt(z)-(1)/(sqrt(z)))^(2)`

Options

A. `(102-46sqrt(3))/(4)`
B. `(102-46sqrt(3))/(2)`
C. `(102-46sqrt(3))/(24)`
D. `(12-46sqrt(3))/(24)`

Answer

To solve the problem where \( z = 6 - 2\sqrt{3} \) and we need to find the value of \( \left( \sqrt{z} - \frac{1}{\sqrt{z}} \right)^2 \), we can follow these steps:

### Step 1: Express the given expression
We start with the expression:
\[ \left( \sqrt{z} - \frac{1}{\sqrt{z}} \right)^2 \]

### Step 2: Use the formula for the square of a binomial
We can expand this expression using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[ \left( \sqrt{z} - \frac{1}{\sqrt{z}} \right)^2 = z + \frac{1}{z} - 2 \]

### Step 3: Substitute the value of \( z \)
Now we substitute \( z = 6 - 2\sqrt{3} \):
\[ z + \frac{1}{z} - 2 \]

### Step 4: Calculate \( \frac{1}{z} \)
To find \( \frac{1}{z} \), we rationalize:
\[ \frac{1}{z} = \frac{1}{6 - 2\sqrt{3}} \cdot \frac{6 + 2\sqrt{3}}{6 + 2\sqrt{3}} = \frac{6 + 2\sqrt{3}}{(6 - 2\sqrt{3})(6 + 2\sqrt{3})} \]
Calculating the denominator:
\[ (6 - 2\sqrt{3})(6 + 2\sqrt{3}) = 36 - (2\sqrt{3})^2 = 36 - 12 = 24 \]
Thus,
\[ \frac{1}{z} = \frac{6 + 2\sqrt{3}}{24} = \frac{1}{4} + \frac{\sqrt{3}}{12} \]

### Step 5: Substitute \( z \) and \( \frac{1}{z} \) back into the expression
Now we have:
\[ z + \frac{1}{z} = (6 - 2\sqrt{3}) + \left( \frac{1}{4} + \frac{\sqrt{3}}{12} \right) \]
Combine these:
\[ = 6 - 2\sqrt{3} + \frac{1}{4} + \frac{\sqrt{3}}{12} \]

### Step 6: Combine the terms
Convert \( 6 \) to a fraction with a common denominator:
\[ 6 = \frac{24}{4} \]
So,
\[ z + \frac{1}{z} = \frac{24}{4} - 2\sqrt{3} + \frac{1}{4} + \frac{\sqrt{3}}{12} \]
Combine the fractions:
\[ = \frac{25}{4} - 2\sqrt{3} + \frac{\sqrt{3}}{12} \]
To combine the square root terms, convert \( -2\sqrt{3} \) to have a common denominator of 12:
\[ -2\sqrt{3} = -\frac{24\sqrt{3}}{12} \]
Thus,
\[ = \frac{25}{4} - \frac{24\sqrt{3}}{12} + \frac{\sqrt{3}}{12} = \frac{25}{4} - \frac{23\sqrt{3}}{12} \]

### Step 7: Final expression
Now we substitute back into our expression:
\[ \left( \sqrt{z} - \frac{1}{\sqrt{z}} \right)^2 = z + \frac{1}{z} - 2 = \left( \frac{25}{4} - \frac{23\sqrt{3}}{12} \right) - 2 \]
Convert \( 2 \) to a fraction:
\[ 2 = \frac{8}{4} \]
Thus, the final expression becomes:
\[ = \frac{25}{4} - \frac{8}{4} - \frac{23\sqrt{3}}{12} = \frac{17}{4} - \frac{23\sqrt{3}}{12} \]

### Final Result
The final answer is:
\[ \frac{17}{4} - \frac{23\sqrt{3}}{12} \]