If `t = (1)/(1-root4(2))` then t equals

Options

A. `(1-root4(2))(2-sqrt(2))`
B. `(1-root4(2))(1+sqrt(2))`
C. `(1+root4(2))(1-sqrt(2))`
D. `-(1+root4(2))(1+sqrt(2))`

Answer

To solve the equation \( t = \frac{1}{1 - 2^{1/4}} \), we will simplify it step by step.

### Step 1: Identify the expression
We start with the expression:
\[ t = \frac{1}{1 - 2^{1/4}} \]

### Step 2: Rationalize the denominator
To simplify the expression, we will rationalize the denominator. We do this by multiplying the numerator and the denominator by the conjugate of the denominator, which is \( 1 + 2^{1/4} \):
\[ t = \frac{1 \cdot (1 + 2^{1/4})}{(1 - 2^{1/4})(1 + 2^{1/4})} \]

### Step 3: Apply the difference of squares
Using the difference of squares formula \( (a - b)(a + b) = a^2 - b^2 \), we can simplify the denominator:
\[ t = \frac{1 + 2^{1/4}}{1^2 - (2^{1/4})^2} \]
\[ t = \frac{1 + 2^{1/4}}{1 - 2^{1/2}} \]

### Step 4: Simplify further
Since \( 2^{1/2} = \sqrt{2} \), we can rewrite the expression:
\[ t = \frac{1 + 2^{1/4}}{1 - \sqrt{2}} \]

### Step 5: Rationalize again
Now we need to rationalize the denominator \( 1 - \sqrt{2} \) by multiplying by its conjugate \( 1 + \sqrt{2} \):
\[ t = \frac{(1 + 2^{1/4})(1 + \sqrt{2})}{(1 - \sqrt{2})(1 + \sqrt{2})} \]

### Step 6: Apply the difference of squares again
Using the difference of squares again for the denominator:
\[ t = \frac{(1 + 2^{1/4})(1 + \sqrt{2})}{1^2 - (\sqrt{2})^2} \]
\[ t = \frac{(1 + 2^{1/4})(1 + \sqrt{2})}{1 - 2} \]
\[ t = \frac{(1 + 2^{1/4})(1 + \sqrt{2})}{-1} \]
\[ t = -(1 + 2^{1/4})(1 + \sqrt{2}) \]

### Step 7: Distribute the negative sign
Now we distribute the negative sign:
\[ t = -1 - 2^{1/4} - \sqrt{2} - 2^{1/4}\sqrt{2} \]

### Step 8: Final simplification
This expression can be left as is or further simplified depending on the context. However, for our purposes, we can conclude here.

### Final Result
Thus, the final value of \( t \) is:
\[ t = -1 - 2^{1/4} - \sqrt{2} - 2^{1/4}\sqrt{2} \]