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. `(15)/( sqrt(x)) - (9)/( sqrt(x)) = (x)^((1)/(2))`
II. ` y^(10) - (36)^(5) = 0 `

Options

A. If ` x gt y `
B. If ` x lt y `
C. If ` x ge Y`
D. If x = y or no relation can be established between 'x' and 'y'

Answer

To solve the given problem, we will analyze and solve the two equations step by step.

### Step 1: Solve Equation I
The first equation is:
\[ \frac{15}{\sqrt{x}} - \frac{9}{\sqrt{x}} = x^{\frac{1}{2}} \]

**Hint for Step 1:** Combine the fractions on the left side.

Combining the fractions:
\[ \frac{15 - 9}{\sqrt{x}} = x^{\frac{1}{2}} \]
This simplifies to:
\[ \frac{6}{\sqrt{x}} = x^{\frac{1}{2}} \]

### Step 2: Cross-Multiply
Now, we can cross-multiply to eliminate the fraction:
\[ 6 = x^{\frac{1}{2}} \cdot \sqrt{x} \]

**Hint for Step 2:** Remember that \(\sqrt{x} = x^{\frac{1}{2}}\).

Since \(\sqrt{x} = x^{\frac{1}{2}}\), we can rewrite the equation as:
\[ 6 = x^{\frac{1}{2}} \cdot x^{\frac{1}{2}} = x^{\frac{1}{2} + \frac{1}{2}} = x^1 \]
Thus, we have:
\[ x = 6 \]

### Step 3: Solve Equation II
The second equation is:
\[ y^{10} - 36^5 = 0 \]

**Hint for Step 3:** Move the term to the other side of the equation.

Rearranging gives:
\[ y^{10} = 36^5 \]

### Step 4: Simplify the Right Side
Now, we can express \(36\) as \(6^2\):
\[ y^{10} = (6^2)^5 \]

**Hint for Step 4:** Use the power of a power property.

Using the property of exponents:
\[ y^{10} = 6^{2 \cdot 5} = 6^{10} \]

### Step 5: Equate the Exponents
Since the bases are the same, we can equate the exponents:
\[ y = 6 \]

### Step 6: Determine the Relation Between x and y
Now we have:
\[ x = 6 \quad \text{and} \quad y = 6 \]

Thus, the relation between \(x\) and \(y\) is:
\[ x = y \]

### Final Answer
The relation between \(x\) and \(y\) is:
\[ \boxed{x = y} \]