Let `alpha, beta` be two roots of the equation `x^(2)+(20)^(1//4)x+ (5)^(1//2)=0`. Then `alpha^(8) + beta^(8)` is equal to

Options

A. 10
B. 100
C. 50
D. 160

Answer

To solve the equation \( x^2 + 20^{1/4}x + \sqrt{5} = 0 \) and find the value of \( \alpha^8 + \beta^8 \) where \( \alpha \) and \( \beta \) are the roots, we can follow these steps:

### Step 1: Identify coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \)
- \( b = 20^{1/4} \)
- \( c = \sqrt{5} \)

### Step 2: Calculate \( \alpha + \beta \) and \( \alpha \beta \)
Using Vieta's formulas:
- \( \alpha + \beta = -\frac{b}{a} = -\frac{20^{1/4}}{1} = -20^{1/4} \)
- \( \alpha \beta = \frac{c}{a} = \frac{\sqrt{5}}{1} = \sqrt{5} \)

### Step 3: Calculate \( \alpha^2 + \beta^2 \)
Using the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \):
\[ \alpha^2 + \beta^2 = (-20^{1/4})^2 - 2\sqrt{5} = 20^{1/2} - 2\sqrt{5} = \sqrt{20} - 2\sqrt{5} \]
\[ = 2\sqrt{5} - 2\sqrt{5} = 0 \]

### Step 4: Calculate \( \alpha^4 + \beta^4 \)
Using the identity \( \alpha^4 + \beta^4 = (\alpha^2 + \beta^2)^2 - 2(\alpha\beta)^2 \):
\[ \alpha^4 + \beta^4 = 0^2 - 2(\sqrt{5})^2 = -2 \cdot 5 = -10 \]

### Step 5: Calculate \( \alpha^8 + \beta^8 \)
Using the identity \( \alpha^8 + \beta^8 = (\alpha^4 + \beta^4)^2 - 2(\alpha^4 \beta^4) \):
\[ \alpha^8 + \beta^8 = (-10)^2 - 2(\sqrt{5})^4 \]
Calculating \( (\sqrt{5})^4 \):
\[ (\sqrt{5})^4 = 25 \]
So,
\[ \alpha^8 + \beta^8 = 100 - 2 \cdot 25 = 100 - 50 = 50 \]

### Final Result
Thus, the value of \( \alpha^8 + \beta^8 \) is \( \boxed{50} \).