Find the value of
`(2^(1//4)- 1) (2^(3//4) + 2^(1//2) +2^(1//4) + 1)`

Answer

To solve the expression \((2^{1/4} - 1)(2^{3/4} + 2^{1/2} + 2^{1/4} + 1)\), we can follow these steps:

### Step 1: Let \( x = 2^{1/4} \)
We start by substituting \( x \) for \( 2^{1/4} \). Thus, we rewrite the expression:
\[ (2^{1/4} - 1)(2^{3/4} + 2^{1/2} + 2^{1/4} + 1) = (x - 1)(x^3 + x^2 + x + 1) \]

### Step 2: Simplify the second part
The expression \( x^3 + x^2 + x + 1 \) can be factored. Notice that:
\[ x^3 + x^2 + x + 1 = (x^3 + 1) + (x^2 + 1) = (x + 1)(x^2 - x + 1) \]
However, we can also directly factor it as:
\[ x^3 + x^2 + x + 1 = (x^2 + 1)(x + 1) \]

### Step 3: Substitute back into the expression
Now we substitute back into the expression:
\[ (x - 1)((x^2 + 1)(x + 1)) \]

### Step 4: Expand the expression
Now we can expand this:
\[ (x - 1)(x^2 + 1)(x + 1) = (x - 1)(x^3 + x^2 + x + 1) \]
Using the identity \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \), we can simplify:
\[ = x^4 - 1 \]

### Step 5: Substitute back \( x = 2^{1/4} \)
Now we substitute back \( x = 2^{1/4} \):
\[ x^4 - 1 = (2^{1/4})^4 - 1 = 2 - 1 = 1 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{1} \]