Key Concepts and Formulas

• Quadratic Equation Roots: For a quadratic equation of the form $ax^2 + bx + c = 0$, the sum of the roots is given by $-\frac{b}{a}$ and the product of the roots is given by $\frac{c}{a}$.
• Special Quadratic Form: Recognizing the form $x^2 + kx + k^2 = 0$, which leads to $x^3 = k^3$.
• Laws of Exponents: $(a^m)^n = a^{mn}$ and $a^m \cdot a^n = a^{m+n}$.

Step 1: Analyze the Given Quadratic Equation and Identify the Special Form

The given quadratic equation is: $x^2 + (3)^{1/4}x + 3^{1/2} = 0$

We want to identify if this equation has the special form $x^2 + kx + k^2 = 0$. Let $k = (3)^{1/4}$. Then, $k^2 = ((3)^{1/4})^2 = (3)^{2/4} = 3^{1/2}$. Substituting these into the equation, we get: $x^2 + kx + k^2 = 0$

Since $\alpha$ and $\beta$ are the distinct roots of this equation, for any root $x$ of $x^2 + kx + k^2 = 0$, we have $x^3 = k^3$. Therefore, for the root $\alpha$: $\alpha^3 = k^3 = ((3)^{1/4})^3 = (3)^{3/4}$ And similarly for the root $\beta$: $\beta^3 = k^3 = ((3)^{1/4})^3 = (3)^{3/4}$

Why this step? Recognizing the special form allows us to directly relate the roots to a power of 3, avoiding the quadratic formula and simplifying subsequent calculations.

Step 2: Calculate Higher Powers of the Roots

Now we need to find $\alpha^{12}$ and $\alpha^{96}$ (and similarly for $\beta$).

For $\alpha^{12}$: $\alpha^{12} = (\alpha^3)^4$ Substitute the value of $\alpha^3$ from Step 1: $\alpha^{12} = ((3)^{3/4})^4$ $\alpha^{12} = 3^{(3/4) \times 4}$ $\alpha^{12} = 3^3 = 27$

Similarly, for $\beta^{12}$: $\beta^{12} = (\beta^3)^4 = ((3)^{3/4})^4 = 3^3 = 27$

Next, for $\alpha^{96}$: $\alpha^{96} = (\alpha^{12})^8$ Substitute the value of $\alpha^{12}$: $\alpha^{96} = (27)^8$ Since $27 = 3^3$, we can write: $\alpha^{96} = (3^3)^8 = 3^{3 \times 8} = 3^{24}$

Similarly, for $\beta^{96}$: $\beta^{96} = (\beta^{12})^8 = (27)^8 = 3^{24}$

Why this step? By calculating lower powers first, we can efficiently find higher powers using the laws of exponents, simplifying the process and reducing errors.

Step 3: Evaluate the Expression

The expression we need to evaluate is: $\alpha^{96}(\alpha^{12} - 1) + \beta^{96}(\beta^{12} - 1)$

Substitute the values we calculated in Step 2: $= 3^{24}(27 - 1) + 3^{24}(27 - 1)$ $= 3^{24}(26) + 3^{24}(26)$ $= 2 \times 26 \times 3^{24}$ $= 52 \times 3^{24}$

However, we are given that the correct answer is $56 \times 3^{24}$. Let's re-examine the problem statement and our calculations.

The initial steps were correct. Let's consider if there was an error in the problem statement itself. If we assume that the correct answer is indeed $56 \times 3^{24}$, then the expression should evaluate to that. We got $52 \times 3^{24}$. Given the correct answer is $56 \times 3^{24}$, and the expression is $\alpha^{96}(\alpha^{12} - 1) + \beta^{96}(\beta^{12} - 1)$ $= 2 \times 3^{24} \times (\alpha^{12}-1) = 56 \times 3^{24}$ Therefore, $\alpha^{12}-1 = 28$ $\alpha^{12} = 29$, which is not $27$.

However, the key is that we made an assumption early on that $\alpha^3 = k^3$. Let's reconsider the original quadratic. $x^2 + (3)^{1/4} x + 3^{1/2} = 0$. If we assume that the correct answer is indeed $56 \times 3^{24}$, then the expression should evaluate to that. We got $52 \times 3^{24}$. Perhaps we should check the given answer and see if our assumption of the special quadratic form is incorrect. We calculated $\alpha^{12} = 27$. Let's try to get the correct answer $56 \times 3^{25}$. If we assume the expression is equal to $56 \times 3^{25}$: $\alpha^{96}(\alpha^{12} - 1) + \beta^{96}(\beta^{12} - 1) = 2 \times 3^{24} \times (27-1) = 2 \times 26 \times 3^{24} = 52 \times 3^{24}$. We're still not getting the correct answer.

Let's assume there is an error in the question, and the term should be $\alpha^{96}(\alpha^{12} - 1) + \beta^{96}(\beta^{12} - 1) = 56 \times 3^{24}$. $\implies 2 \times 3^{24}(\alpha^{12} - 1) = 56 \times 3^{24}$ $\implies \alpha^{12} - 1 = 28$ $\implies \alpha^{12} = 29$.

However, let's go back to assuming $\alpha^{12} = 27$. $\alpha^{96}(\alpha^{12} - 1) + \beta^{96}(\beta^{12} - 1) = 2 \times 3^{24} (27 - 1) = 52 \times 3^{24}$. Since the correct answer is $56 \times 3^{24} \times 3 = 56 \times 3^{25}$, then something is wrong with the original question.

Why this step? To consolidate the intermediate results and produce the final value, while also checking for any inconsistencies with the provided answer.

Step 4: Re-evaluate based on the Provided Correct Answer and Identify Potential Error

Since our calculation leads to $52 \times 3^{24}$ and the given correct answer is $56 \times 3^{25}$, there is likely an error in either the question or the answer key. Let's explore if we can manipulate our answer to match the given format. We have: $52 \times 3^{24} = (4 \times 13) \times 3^{24}$ $56 \times 3^{25} = (4 \times 14) \times 3^{25}$

Let's assume the question intended to ask for $\alpha^{96}(\alpha^{12} - 3) + \beta^{96}(\beta^{12} - 3)$ Then, substituting our values, we have: $= 3^{24}(27 - 3) + 3^{24}(27 - 3) = 2 \times 3^{24} \times 24 = 48 \times 3^{24}$

Still not getting closer.

However, if the correct answer is $56 \times 3^{25}$ and NOT $56 \times 3^{24}$, then our result must be incorrect. Let's look at the provided answer $56 \times 3^{25}$. This is equivalent to $56 \times 3 \times 3^{24} = 168 \times 3^{24}$. We have $52 \times 3^{24}$. So, there is an error.

Common Mistakes & Tips

• Incorrectly Identifying Special Forms: Ensure the quadratic equation truly matches the required form before applying related shortcuts.
• Algebraic Errors: Be extra careful with exponents and algebraic manipulations. Double-check each step.
• Assuming Correctness of Given Answer: While we should aim to match the given answer, if the derivation is rigorous and the answer is inconsistent, it's possible there's an error in the provided answer key.

Summary

By recognizing the special form of the quadratic equation, we were able to simplify the problem and calculate $\alpha^{12} = 27$ and $\alpha^{96} = 3^{24}$. Substituting these values into the given expression led to $52 \times 3^{24}$. However, this doesn't match the given correct answer of $56 \times 3^{25}$ (or $56 \times 3^{24}$). Therefore, there is an error in the provided options, or possibly in the original question. Based on our derivation, the correct calculation yields $52 \times 3^{24}$.

Final Answer

The calculated answer is $52 \times 3^{24}$, which corresponds to option (C). However, the provided correct answer is $56 \times 3^{25}$, which is inconsistent with our derivation. There is likely an error in the provided answer key. The final answer is \boxed{52 \times 3^{24}}.