Key Concepts and Formulas

• Roots of a Quadratic Equation: If $\alpha$ and $\beta$ are roots of the quadratic equation $ax^2 + bx + c = 0$, then $\alpha$ and $\beta$ satisfy the equation: $a\alpha^2 + b\alpha + c = 0$ and $a\beta^2 + b\beta + c = 0$.
• Algebraic Manipulation: Strategic rearrangement and squaring of equations can simplify expressions and reveal relationships between roots and coefficients.
• Exponent Rules: $(a^m)^n = a^{mn}$ and $a^{m/n} = \sqrt[n]{a^m}$

Step-by-Step Solution

Step 1: Rewrite the Given Equation and Isolate Terms

We begin with the given equation: $x^2 + (20)^{1/4}x + (5)^{1/2} = 0$ We rewrite the fractional exponents as radicals: $x^2 + \sqrt[4]{20}x + \sqrt{5} = 0$ Our goal is to isolate the term with x to prepare for squaring. Rearranging, we get: $x^2 + \sqrt{5} = -\sqrt[4]{20}x$ Why? Isolating the $x$ term will allow us to eliminate it when we square both sides, leading to a simpler equation involving higher powers of $x$.

Step 2: Square Both Sides of the Equation

Now, we square both sides of the equation: $(x^2 + \sqrt{5})^2 = (-\sqrt[4]{20}x)^2$ Why? Squaring eliminates the fourth root on the right side and introduces $x^4$ on the left, getting us closer to $x^8$.

Expanding the left side using $(a+b)^2 = a^2 + 2ab + b^2$, we have: $x^4 + 2\sqrt{5}x^2 + 5 = (\sqrt[4]{20})^2 x^2$ $x^4 + 2\sqrt{5}x^2 + 5 = \sqrt{20}x^2$

Step 3: Simplify the Equation

We simplify $\sqrt{20}$: $\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}$ Why? Simplifying radicals makes it easier to identify and combine like terms.

Substituting this back into the equation: $x^4 + 2\sqrt{5}x^2 + 5 = 2\sqrt{5}x^2$ Subtracting $2\sqrt{5}x^2$ from both sides, we get: $x^4 + 5 = 0$ Why? This significantly simplifies the equation, eliminating the $x^2$ term and leading to a direct relationship between $x^4$ and a constant.

Step 4: Determine $x^8$

From the simplified equation $x^4 + 5 = 0$, we isolate $x^4$: $x^4 = -5$ Why? This provides a direct value for $x^4$, which we can then use to find $x^8$.

To find $x^8$, we square both sides of the equation: $(x^4)^2 = (-5)^2$ $x^8 = 25$ Why? This gives us the value of $x^8$ directly.

Step 5: Calculate $\alpha^8 + \beta^8$

Since $\alpha$ and $\beta$ are roots of the original equation, they must satisfy the derived equation $x^8 = 25$. Therefore: $\alpha^8 = 25$ $\beta^8 = 25$ Why? Because any root of the original equation will satisfy any equivalent form or derived equation.

Finally, we calculate the sum: $\alpha^8 + \beta^8 = 25 + 25 = 50$

Common Mistakes & Tips

• Radical Simplification: Always simplify radicals to make calculations easier and avoid errors.
• Squaring Carefully: Ensure correct expansion when squaring binomials, especially when dealing with radicals.
• Avoid Direct Root Calculation: For problems asking for powers of roots, try to manipulate the equation instead of finding the roots directly.

Summary

We started with a quadratic equation and strategically manipulated it by isolating terms, squaring, and simplifying radicals. This led us to a simple equation, $x^8 = 25$. Since both roots $\alpha$ and $\beta$ must satisfy this equation, we found that $\alpha^8 + \beta^8 = 25 + 25 = 50$.

The final answer is \boxed{50}, which corresponds to option (C).