I. `x = sqrt([(36)^(1//2) xx(1296)^(1//4)])`
II.` 2y+3z = 33`
III. ` 6y+5z = 71`

Options

A. `x lt y = z `
B. ` x le y lt z `
C. ` x lt y gt z `
D. ` x = y gt z `

Explanation

To solve the given equations step by step, we will first tackle the equation for \( x \) and then solve the system of equations for \( y \) and \( z \).

### Step 1: Solve for \( x \)

We start with the equation:
\[ x = \sqrt{(36)^{1/2} \times (1296)^{1/4}} \]

**Calculate \( (36)^{1/2} \)**:
\[ (36)^{1/2} = \sqrt{36} = 6 \]

**Calculate \( (1296)^{1/4} \)**:
To find \( (1296)^{1/4} \), we can first find the square root of \( 1296 \):
\[ \sqrt{1296} = 36 \quad \text{(since \( 36 \times 36 = 1296 \))} \]
Now, take the square root of \( 36 \):
\[ (36)^{1/2} = 6 \]

Thus, \( (1296)^{1/4} = 6 \).

**Combine the results**:
Now substitute back into the equation for \( x \):
\[ x = \sqrt{6 \times 6} = \sqrt{36} = 6 \]

### Step 2: Solve the system of equations for \( y \) and \( z \)

We have two equations:
1. \( 2y + 3z = 33 \) (Equation 1)
2. \( 6y + 5z = 71 \) (Equation 2)

**Multiply Equation 1 by 3**:
\[ 3(2y + 3z) = 3(33) \implies 6y + 9z = 99 \quad \text{(Equation 3)} \]

**Subtract Equation 2 from Equation 3**:
\[ (6y + 9z) - (6y + 5z) = 99 - 71 \]
This simplifies to:
\[ 4z = 28 \implies z = 7 \]

### Step 3: Substitute \( z \) back to find \( y \)

Substituting \( z = 7 \) into Equation 1:
\[ 2y + 3(7) = 33 \implies 2y + 21 = 33 \]
Subtract 21 from both sides:
\[ 2y = 12 \implies y = 6 \]

### Summary of values
Now we have:
- \( x = 6 \)
- \( y = 6 \)
- \( z = 7 \)

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

We compare the values:
- \( x = 6 \)
- \( y = 6 \)
- \( z = 7 \)

From this, we can conclude:
- \( x = y \)
- \( y < z \)

### Final Result
The relationship is:
\[ x \leq y < z \]