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 problem \( t = \frac{1}{1 - \sqrt[4]{2}} \), we will go through the following steps:

### Step 1: Rewrite the Expression
We start with the expression for \( t \):
\[ t = \frac{1}{1 - \sqrt[4]{2}} \]
Here, \( \sqrt[4]{2} \) can be expressed as \( 2^{1/4} \).

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

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

### Step 4: Rationalize Again
Next, we need to rationalize the new denominator \( 1 - \sqrt{2} \). We multiply the numerator and denominator by the conjugate \( 1 + \sqrt{2} \):
\[ t = \frac{(1 + \sqrt[4]{2})(1 + \sqrt{2})}{(1 - \sqrt{2})(1 + \sqrt{2})} \]

### Step 5: Simplify the Denominator
Again applying the difference of squares:
\[ t = \frac{(1 + \sqrt[4]{2})(1 + \sqrt{2})}{1^2 - (\sqrt{2})^2} \]
Calculating \( (\sqrt{2})^2 \) gives us \( 2 \):
\[ t = \frac{(1 + \sqrt[4]{2})(1 + \sqrt{2})}{1 - 2} \]
This simplifies to:
\[ t = \frac{(1 + \sqrt[4]{2})(1 + \sqrt{2})}{-1} \]
Thus:
\[ t = -(1 + \sqrt[4]{2})(1 + \sqrt{2}) \]

### Final Expression
The final expression for \( t \) is:
\[ t = -(1 + \sqrt[4]{2})(1 + \sqrt{2}) \]