If `xsqrt(243) = ysqrt(867)`, where `x` and `y` are co-prime numbers, then the value of `(x-y)` is

Options

A. 3
B. 8
C. 4
D. 6

Answer

To solve the equation \( x \sqrt{243} = y \sqrt{867} \) where \( x \) and \( y \) are co-prime numbers, we will follow these steps:

### Step 1: Simplify the square roots
First, we need to simplify \( \sqrt{243} \) and \( \sqrt{867} \).

- **Calculating \( \sqrt{243} \)**:
\[ 243 = 3^5 \quad \text{(since } 3 \times 3 \times 3 \times 3 \times 3 = 243\text{)} \]
Therefore,
\[ \sqrt{243} = \sqrt{3^5} = 3^{5/2} = 3^2 \cdot \sqrt{3} = 9\sqrt{3} \]

- **Calculating \( \sqrt{867} \)**:
\[ 867 = 3 \times 289 = 3 \times 17^2 \quad \text{(since } 17 \times 17 = 289\text{)} \]
Therefore,
\[ \sqrt{867} = \sqrt{3 \times 17^2} = \sqrt{3} \cdot 17 = 17\sqrt{3} \]

### Step 2: Substitute back into the equation
Now, substituting these values back into the equation:
\[ x \cdot 9\sqrt{3} = y \cdot 17\sqrt{3} \]

### Step 3: Cancel out \( \sqrt{3} \)
Since \( \sqrt{3} \) is common on both sides, we can cancel it:
\[ x \cdot 9 = y \cdot 17 \]

### Step 4: Rearranging the equation
Rearranging gives us:
\[ \frac{x}{y} = \frac{17}{9} \]

### Step 5: Assign values to \( x \) and \( y \)
Since \( x \) and \( y \) are co-prime, we can assign:
\[ x = 17 \quad \text{and} \quad y = 9 \]

### Step 6: Calculate \( x - y \)
Now, we can find \( x - y \):
\[ x - y = 17 - 9 = 8 \]

### Final Answer
The value of \( x - y \) is \( 8 \).

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