Calculate 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 calculate the value of the expression \(((5^{(1/4)} - 1)(5^{(3/4)} + 5^{(1/2)} + 5^{(1/4)} + 1))\), we can simplify it step by step.

### Step-by-Step Solution:

1. **Identify the Expression**:
We start with the expression:
\[ (5^{(1/4)} - 1)(5^{(3/4)} + 5^{(1/2)} + 5^{(1/4)} + 1) \]

2. **Rewrite \(5^{(3/4)}\)**:
Notice that \(5^{(3/4)}\) can be rewritten as:
\[ 5^{(3/4)} = 5^{(1/2)} \cdot 5^{(1/4)} \]
So, we can express the second part of the expression as:
\[ 5^{(3/4)} + 5^{(1/2)} + 5^{(1/4)} + 1 = 5^{(1/2)} \cdot 5^{(1/4)} + 5^{(1/2)} + 5^{(1/4)} + 1 \]

3. **Factor Out Common Terms**:
We can factor out \(5^{(1/4)}\) from the terms:
\[ = 5^{(1/4)}(5^{(1/2)} + 1) + 5^{(1/4)} + 1 \]
This simplifies to:
\[ = 5^{(1/4)}(5^{(1/2)} + 1 + 1) + 1 = 5^{(1/4)}(5^{(1/2)} + 2) + 1 \]

4. **Combine the Terms**:
Now, we can combine the terms:
\[ (5^{(1/4)} - 1)(5^{(1/4)}(5^{(1/2)} + 2) + 1) \]

5. **Use the Difference of Squares**:
The expression \((A - B)(A + B) = A^2 - B^2\) can be applied here. Let:
\[ A = 5^{(1/4)} + 1 \quad \text{and} \quad B = 1 \]
Thus, we have:
\[ (5^{(1/4)} - 1)(5^{(1/4)} + 1) = (5^{(1/4)})^2 - 1^2 \]

6. **Calculate \(A^2 - B^2\)**:
Now, calculate:
\[ (5^{(1/4)})^2 - 1 = 5^{(1/2)} - 1 \]

7. **Final Calculation**:
The expression simplifies to:
\[ 5^{(1/2)} - 1 = \sqrt{5} - 1 \]

### Final Answer:
The value of the expression is:
\[ \sqrt{5} - 1 \]