Two equations (I) and (II) are given in each question. On the basis of these equations you have to decide the relation between x and y and give answer
I. ` x = 3 sqrt(2197) `
II. ` y^(2) = 169`

Options

A. If ` x gt y `
B. If ` x lt y `
C. If ` x ge Y`
D. If ` x le y `

Answer

To solve the given problem, we need to analyze the two equations provided:

### Given Equations:
1. \( I. \quad x = 3 \sqrt{2197} \)
2. \( II. \quad y^2 = 169 \)

### Step 1: Calculate the value of \( x \)

First, we need to simplify the expression for \( x \):

\[ x = 3 \sqrt{2197} \]

Now, we need to find the square root of \( 2197 \). We can factor \( 2197 \) to find its square root.

Since \( 2197 = 13^3 \), we can express it as:

\[ \sqrt{2197} = \sqrt{13^3} = 13 \sqrt{13} \]

Now substituting back into the equation for \( x \):

\[ x = 3 \cdot 13 \sqrt{13} = 39 \sqrt{13} \]

### Step 2: Calculate the value of \( y \)

Next, we solve the second equation:

\[ y^2 = 169 \]

Taking the square root of both sides gives us:

\[ y = \pm \sqrt{169} = \pm 13 \]

This means \( y \) can be either \( 13 \) or \( -13 \).

### Step 3: Compare \( x \) and \( y \)

Now we have:

- \( x = 39 \sqrt{13} \)
- \( y = 13 \) or \( y = -13 \)

Next, we need to find the approximate value of \( 39 \sqrt{13} \):

Calculating \( \sqrt{13} \) (approximately \( 3.605 \)):

\[ x \approx 39 \cdot 3.605 \approx 140.595 \]

### Step 4: Determine the relationship between \( x \) and \( y \)

Now we compare \( x \) with the possible values of \( y \):

1. When \( y = 13 \):
- \( 140.595 > 13 \)

2. When \( y = -13 \):
- \( 140.595 > -13 \)

In both cases, \( x \) is greater than \( y \).

### Conclusion

Thus, we conclude that:

\[ x > y \]

### Final Answer

The relation between \( x \) and \( y \) is \( x > y \).

---