https://file.aiquickdraw.com/mathbot/file/1727314274980sfm9m7pk.jpg

The image shows two mathematical expressions involving square roots that need to be simplified. Let's solve each part one by one.

### a. \( \frac{2\sqrt{3}}{\sqrt{6}} \)

We can simplify this by dividing the terms inside the square roots first:

\[
\frac{2\sqrt{3}}{\sqrt{6}} = \frac{2\sqrt{3}}{\sqrt{2 \times 3}} = \frac{2\sqrt{3}}{\sqrt{2} \times \sqrt{3}} = \frac{2\sqrt{3}}{\sqrt{3} \times \sqrt{2}}
\]

Now, cancel out the \( \sqrt{3} \):

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

To rationalize the denominator, multiply by \( \frac{\sqrt{2}}{\sqrt{2}} \):

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

Thus, the simplified form is:

\[
\boxed{\sqrt{2}}
\]

### b. \( \frac{\sqrt{2}}{\sqrt{5} - \sqrt{3}} \)

We need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of \( \sqrt{5} - \sqrt{3} \), which is \( \sqrt{5} + \sqrt{3} \):

\[
\frac{\sqrt{2}}{\sqrt{5} - \sqrt{3}} \times \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}} = \frac{\sqrt{2}(\sqrt{5} + \sqrt{3})}{(\sqrt{5})^2 - (\sqrt{3})^2}
\]

Now simplify the denominator:

\[
= \frac{\sqrt{2}(\sqrt{5} + \sqrt{3})}{5 - 3} = \frac{\sqrt{2}(\sqrt{5} + \sqrt{3})}{2}
\]

Thus, the simplified form is:

\[
\boxed{\frac{\sqrt{2}(\sqrt{5} + \sqrt{3})}{2}}
\]

---

Would you like any further explanation on these steps or have any additional questions?

### Related Questions:
1. How do you simplify expressions involving multiple square roots?
2. What is the method of rationalizing denominators?
3. Can you explain how to multiply conjugates and their significance in simplifying?
4. How do we apply the distributive property when simplifying square roots?
5. What other types of expressions benefit from rationalizing the denominator?

### Tip:
Always check whether the square roots in the numerator and denominator can be simplified further before rationalizing the denominator.

Simplify the following radical expressions: a. (2√3) / √6, b. (√2) / (√5 - √3)

Step-by-step guide to simplifying radical expressions like (2√3) / √6 and (√2) / (√5 - √3). Learn how to rationalize denominators and use conjugates to simplify square roots.

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