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The expression given in the image is \(\frac{\sqrt{45}}{\sqrt{10}}\). To simplify this expression, we can use the property of radicals that states:

\[
\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}
\]

### Step-by-Step Simplification

1. **Combine the Radicals:**

   \[
   \frac{\sqrt{45}}{\sqrt{10}} = \sqrt{\frac{45}{10}}
   \]

2. **Simplify the Fraction Inside the Radical:**

   \[
   \frac{45}{10} = \frac{9 \cdot 5}{2 \cdot 5} = \frac{9}{2}
   \]

   So we have:

   \[
   \sqrt{\frac{45}{10}} = \sqrt{\frac{9}{2}}
   \]

3. **Separate the Radicals Using Properties:**

   \[
   \sqrt{\frac{9}{2}} = \sqrt{9} \cdot \sqrt{\frac{1}{2}}
   \]

4. **Calculate the Square Roots:**

   - \(\sqrt{9} = 3\)

   Therefore:

   \[
   \sqrt{\frac{9}{2}} = 3 \cdot \sqrt{\frac{1}{2}}
   \]

5. **Rationalize the Denominator:**

   To further simplify \(\sqrt{\frac{1}{2}}\), we rationalize it:

   \[
   \sqrt{\frac{1}{2}} = \frac{\sqrt{1}}{\sqrt{2}} = \frac{1}{\sqrt{2}}
   \]

   Multiply numerator and denominator by \(\sqrt{2}\):

   \[
   \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}
   \]

6. **Final Simplification:**

   Substitute back into the expression:

   \[
   3 \cdot \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2}
   \]

### Simplified Expression

The simplified form of \(\frac{\sqrt{45}}{\sqrt{10}}\) is \(\frac{3\sqrt{2}}{2}\).

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Would you like to see more details, or do you have any questions? Here are some related questions you might find useful:

1. What are the properties of radicals that allow combining and simplifying them?
2. How do you simplify a square root of a fraction?
3. What does it mean to rationalize the denominator?
4. Why do we need to rationalize the denominator, and when is it necessary?
5. How do you simplify expressions with higher roots, such as cube roots?
6. What are some common mistakes to avoid when simplifying radical expressions?
7. Can these methods be used to simplify radical expressions involving variables?
8. How can you check if your simplified expression is correct?

**Tip:** When simplifying radicals, always look for perfect squares (or cubes, etc.) within the radicand to simplify expressions quickly.

Learn how to simplify the expression √45 / √10 step-by-step. This problem involves radicals, simplifying fractions, and rationalizing denominators, suitable for high school students.

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