If `alpha,beta` are root of `x^2+20^(1/4)x+sqrt5=0` then find the value of `alpha^8+beta^8`

Answer

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

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

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

### Step 3: Calculate \( \alpha^2 + \beta^2 \)
We can find \( \alpha^2 + \beta^2 \) using the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values we found:
\[ \alpha^2 + \beta^2 = (-20^{1/4})^2 - 2\sqrt{5} \]
Calculating this gives:
\[ \alpha^2 + \beta^2 = 20^{1/2} - 2\sqrt{5} = \sqrt{20} - 2\sqrt{5} = 2\sqrt{5} - 2\sqrt{5} = 0 \]

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

### Step 5: Calculate \( \alpha^8 + \beta^8 \)
Using the identity:
\[ \alpha^8 + \beta^8 = (\alpha^4 + \beta^4)^2 - 2(\alpha^4 \beta^4) \]
We need \( \alpha^4 \beta^4 = (\alpha \beta)^4 = (\sqrt{5})^4 = 25 \).
Now substituting:
\[ \alpha^8 + \beta^8 = (-10)^2 - 2 \cdot 25 = 100 - 50 = 50 \]

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