The problem provided in the image is as follows:

$\frac{-\sqrt{10}}{3\sqrt{18}}$

Simplification:

1. Simplify the denominator:

   • $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$.
   • So, $3\sqrt{18} = 3 \cdot 3\sqrt{2} = 9\sqrt{2}$.

   The expression now becomes: $\frac{-\sqrt{10}}{9\sqrt{2}}.$

2. Rationalize the denominator: Multiply the numerator and denominator by $\sqrt{2}$ to eliminate the square root in the denominator: $\frac{-\sqrt{10}}{9\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{-\sqrt{10} \cdot \sqrt{2}}{9 \cdot 2}.$

3. Simplify further:

   • $\sqrt{10} \cdot \sqrt{2} = \sqrt{20}$,
   • $\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}$.

   So the numerator becomes $-2\sqrt{5}$, and the denominator is $18$: $\frac{-2\sqrt{5}}{18}.$

4. Simplify the fraction: $\frac{-2\sqrt{5}}{18} = \frac{-\sqrt{5}}{9}.$

Final Answer:

$\boxed{\frac{-\sqrt{5}}{9}}$

Would you like additional details or explanations?

Related Questions:

1. How is the square root of a product ($\sqrt{ab}$) simplified?
2. What is the process for rationalizing a denominator involving square roots?
3. How does simplifying fractions with radicals work?
4. Why does $\sqrt{18}$ simplify to $3\sqrt{2}$?
5. What are other examples of rationalizing denominators?

Tip:

Always check for perfect squares in the radicand when simplifying square roots.