In each of the following questions, two equations (I) and (II) are given . Solve the equations and mark the correct options
I. ` x^(2) = 4`
II. ` y^(5) = 32 `

Options

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

Answer

To solve the given equations and determine the relationship between \( x \) and \( y \), we will follow these steps:

### Step 1: Solve the first equation \( x^2 = 4 \)

To solve for \( x \), we take the square root of both sides:

\[ x = \pm \sqrt{4} \]

Calculating the square root gives us:

\[ x = \pm 2 \]

So, the possible values for \( x \) are:

\[ x = 2 \quad \text{or} \quad x = -2 \]

### Step 2: Solve the second equation \( y^5 = 32 \)

To solve for \( y \), we can express 32 as a power of 2:

\[ 32 = 2^5 \]

Thus, we can rewrite the equation as:

\[ y^5 = 2^5 \]

Taking the fifth root of both sides gives us:

\[ y = 2 \]

### Step 3: Compare the values of \( x \) and \( y \)

Now we have two values for \( x \) and one value for \( y \):

- Possible values of \( x \): \( 2 \) and \( -2 \)
- Value of \( y \): \( 2 \)

Now we will compare:

1. When \( x = 2 \):
\[ x = 2 \quad \text{and} \quad y = 2 \quad \Rightarrow \quad x = y \]

2. When \( x = -2 \):
\[ x = -2 \quad \text{and} \quad y = 2 \quad \Rightarrow \quad x < y \]

### Conclusion

From our comparisons, we find that:

- In one case, \( x \) is equal to \( y \).
- In another case, \( x \) is less than \( y \).

Thus, we can conclude that the relationship between \( x \) and \( y \) is:

\[ x \leq y \]

### Final Answer

The correct option is \( x \leq y \).

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