Calulate the value of `(5^(1//4)-1) (5^(3//4)+5^(1//2)+5^(1//4)+1)`

Options

A. 5
B. 4
C. 10
D. 25

Answer

To solve the expression \((5^{1/4} - 1)(5^{3/4} + 5^{1/2} + 5^{1/4} + 1)\), we will follow these steps:

### Step 1: Simplify the Expression
We can start by letting \(x = 5^{1/4}\). Therefore, we can rewrite the expression as:
\[ (x - 1)(x^3 + x^2 + x + 1) \]

### Step 2: Factor the Polynomial
The polynomial \(x^3 + x^2 + x + 1\) can be factored. We can group the terms:
\[ x^3 + x^2 + x + 1 = (x^3 + 1) + (x^2 + 1) = (x + 1)(x^2 - x + 1) \]
However, a more straightforward way to factor it is to recognize that:
\[ x^3 + x^2 + x + 1 = (x^2 + 1)(x + 1) \]

### Step 3: Substitute Back
Now substituting back \(x = 5^{1/4}\):
\[ (5^{1/4} - 1)((5^{1/4})^3 + (5^{1/4})^2 + (5^{1/4}) + 1) \]
This simplifies to:
\[ (5^{1/4} - 1)(5^{3/4} + 5^{2/4} + 5^{1/4} + 1) \]

### Step 4: Calculate the Value
Now we can calculate the value of \((5^{1/4} - 1)(5^{3/4} + 5^{1/2} + 5^{1/4} + 1)\):
1. \(5^{1/4} - 1\) is a small positive number.
2. The second part \(5^{3/4} + 5^{1/2} + 5^{1/4} + 1\) is a sum of positive numbers, which is larger than 1.

### Step 5: Evaluate the Expression
Now we can evaluate the expression:
\[ (5^{1/4} - 1)(5^{3/4} + 5^{1/2} + 5^{1/4} + 1) = (5^{1/4} - 1)(5^{3/4} + 5^{1/2} + 5^{1/4} + 1) \]
This can be computed directly, but we can also recognize that the expression simplifies to:
\[ 5 - 1 = 4 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{4} \]