You have the equation \( \frac{3x^4 - 6}{2y} = 0 \). Let's solve it step by step.

### Step 1: Isolate the numerator.
Since the denominator is \(2y\), we need to focus on the numerator:
\[
3x^4 - 6 = 0
\]
Solve for \(x\).

### Step 2: Solve \(3x^4 - 6 = 0\).

1. Add 6 to both sides:
   \[
   3x^4 = 6
   \]

2. Divide both sides by 3:
   \[
   x^4 = 2
   \]

3. Take the fourth root of both sides:
   \[
   x = \pm \sqrt[4]{2}
   \]

### Final Answer:
\[
x = \pm \sqrt[4]{2}
\]

### Step 3: Implications on the original equation.
Since the numerator has been solved, the entire expression \(\frac{3x^4 - 6}{2y} = 0\) holds true for \(x = \pm \sqrt[4]{2}\) regardless of the value of \(y\), as long as \(y \neq 0\) (to avoid division by zero).

Would you like more details or have any questions?

Here are 5 related questions:
1. What are the properties of even roots, such as the fourth root?
2. How does division by zero affect equations?
3. Can you solve equations of the form \(ax^n + b = 0\) for other values of \(n\)?
4. What are the steps to simplify fractions with variables in the denominator?
5. How does solving for roots differ when working with complex numbers?

**Tip:** Always check if variables in denominators lead to undefined expressions by ensuring the denominator is non-zero.

Explore the step-by-step solution to solving the polynomial equation 3x^4 - 6 = 0 divided by 2y. Understand key algebraic concepts including the properties of even roots and solving for x in higher-order equations. Suitable for students in Grades 10-12.

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