` x^(4) - 227 = 398, y^(2) + 321 = 346`

Options

A. if ` x gt y`
B. if ` x lt y`
C. if ` x ge y`
D. if x = y or relation cannot be established between 'x' and 'y'.

Explanation

To solve the equations \( x^4 - 227 = 398 \) and \( y^2 + 321 = 346 \), we will follow these steps:

### Step 1: Solve the first equation for \( x \)

Start with the equation:
\[ x^4 - 227 = 398 \]

Add 227 to both sides:
\[ x^4 = 398 + 227 \]
\[ x^4 = 625 \]

### Step 2: Find \( x \)

Now, take the fourth root of both sides:
\[ x = \sqrt[4]{625} \]

Since \( 625 = 25^2 \), we can rewrite it as:
\[ x = \sqrt[4]{25^2} = \sqrt{25} = 5 \]

Thus, we have:
\[ x = 5 \]

### Step 3: Solve the second equation for \( y \)

Now, move to the second equation:
\[ y^2 + 321 = 346 \]

Subtract 321 from both sides:
\[ y^2 = 346 - 321 \]
\[ y^2 = 25 \]

### Step 4: Find \( y \)

Now, take the square root of both sides:
\[ y = \pm \sqrt{25} \]
\[ y = \pm 5 \]

### Conclusion

We have found:
\[ x = 5 \quad \text{and} \quad y = 5 \text{ or } y = -5 \]

### Relation between \( x \) and \( y \)

The values of \( x \) and \( y \) show that \( x = y \) when \( y = 5 \), and there is no relation when \( y = -5 \).