If `x=sqrt(6) and y^(2)=12`, then `(4)/(xy)=`

Options

A. `(3)/(2sqrt(2))`
B. `(sqrt(2))/(3)`
C. `(3)/(sqrt(2))`
D. `(2sqrt(2))/(3)`

Answer

To solve the problem step by step, we need to find the value of \( \frac{4}{xy} \) given that \( x = \sqrt{6} \) and \( y^2 = 12 \).

### Step 1: Determine the value of \( y \)
We know that \( y^2 = 12 \). To find \( y \), we take the square root of both sides:
\[ y = \sqrt{12} \]
We can simplify \( \sqrt{12} \):
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]
Thus, \( y = 2\sqrt{3} \) or \( y = -2\sqrt{3} \) (since the square root can be positive or negative).

### Step 2: Substitute the values of \( x \) and \( y \) into \( xy \)
Now, we substitute the values of \( x \) and \( y \):
\[ x = \sqrt{6} \quad \text{and} \quad y = 2\sqrt{3} \]
Calculating \( xy \):
\[ xy = \sqrt{6} \times 2\sqrt{3} = 2 \times \sqrt{6} \times \sqrt{3} = 2 \times \sqrt{18} = 2 \times 3\sqrt{2} = 6\sqrt{2} \]

### Step 3: Calculate \( \frac{4}{xy} \)
Now we can find \( \frac{4}{xy} \):
\[ \frac{4}{xy} = \frac{4}{6\sqrt{2}} = \frac{2}{3\sqrt{2}} \]

### Step 4: Rationalize the denominator
To simplify \( \frac{2}{3\sqrt{2}} \), we can rationalize the denominator:
\[ \frac{2}{3\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{3 \times 2} = \frac{\sqrt{2}}{3} \]

### Final Answer
Thus, the value of \( \frac{4}{xy} \) is:
\[ \frac{\sqrt{2}}{3} \]