To simplify the expression \( \sqrt{\frac{1}{2}} \times \sqrt{\frac{5}{6}} \), you can use the property of square roots that states \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \).
First, combine the fractions under one square root:
\[ \sqrt{\frac{1}{2}} \times \sqrt{\frac{5}{6}} = \sqrt{\frac{1 \cdot 5}{2 \cdot 6}} = \sqrt{\frac{5}{12}} \]
Now, \( \sqrt{\frac{5}{12}} \) can be simplified further:
- Split the square root:
\[ \sqrt{\frac{5}{12}} = \frac{\sqrt{5}}{\sqrt{12}} \]
- Simplify \( \sqrt{12} \):
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]
- Substitute back:
\[ \frac{\sqrt{5}}{\sqrt{12}} = \frac{\sqrt{5}}{2\sqrt{3}} \]
So, the final result is:
\[ \sqrt{\frac{1}{2}} \times \sqrt{\frac{5}{6}} = \frac{\sqrt{5}}{2\sqrt{3}} \]
This is the simplified form of the original expression.