Simplify the following radical expressions.

It looks like you've uploaded an image containing a list of expressions that need to be rationalized — that is, simplified so that the denominator does not contain a radical (square root). Let's go through each one step by step and rationalize them.

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(i) $\frac{2}{\sqrt{3}}$

To rationalize, multiply numerator and denominator by $\sqrt{3}$:

$$
\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
$$

✔ Answer: $\boxed{\frac{2\sqrt{3}}{3}}$

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(ii) $\frac{3}{\sqrt{10}}$

Multiply numerator and denominator by $\sqrt{10}$:

$$
\frac{3}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{3\sqrt{10}}{10}
$$

✔ Answer: $\boxed{\frac{3\sqrt{10}}{10}}$

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(iii) $\frac{\sqrt{5} + 1}{\sqrt{2}}$

Multiply numerator and denominator by $\sqrt{2}$:

$$
\frac{\sqrt{5} + 1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{(\sqrt{5} + 1)\sqrt{2}}{2} = \frac{\sqrt{10} + \sqrt{2}}{2}
$$

✔ Answer: $\boxed{\frac{\sqrt{10} + \sqrt{2}}{2}}$

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(iv) $\frac{\sqrt{10} + \sqrt{15}}{\sqrt{2}}$

Multiply numerator and denominator by $\sqrt{2}$:

$$
\frac{\sqrt{10} + \sqrt{15}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{(\sqrt{10} + \sqrt{15})\sqrt{2}}{2}
$$

Now simplify each term:

- $\sqrt{10} \cdot \sqrt{2} = \sqrt{20} = 2\sqrt{5}$
- $\sqrt{15} \cdot \sqrt{2} = \sqrt{30}$

So:

$$
= \frac{2\sqrt{5} + \sqrt{30}}{2}
$$

✔ Answer: $\boxed{\frac{2\sqrt{5} + \sqrt{30}}{2}}$

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(v) $\frac{2 + \sqrt{3}}{3}$

This expression already has a rational denominator (3), so no rationalization is needed.

✔ Answer: $\boxed{\frac{2 + \sqrt{3}}{3}}$ (already simplified)

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(vi) $\frac{\sqrt{2} - 1}{\sqrt{5}}$

Multiply numerator and denominator by $\sqrt{5}$:

$$
\frac{\sqrt{2} - 1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{(\sqrt{2} - 1)\sqrt{5}}{5} = \frac{\sqrt{10} - \sqrt{5}}{5}
$$

✔ Answer: $\boxed{\frac{\sqrt{10} - \sqrt{5}}{5}}$

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✔ Final Answers:

| Expression | Rationalized Form |
|----------|-------------------|
| (i) $\frac{2}{\sqrt{3}}$ | $\frac{2\sqrt{3}}{3}$ |
| (ii) $\frac{3}{\sqrt{10}}$ | $\frac{3\sqrt{10}}{10}$ |
| (iii) $\frac{\sqrt{5} + 1}{\sqrt{2}}$ | $\frac{\sqrt{10} + \sqrt{2}}{2}$ |
| (iv) $\frac{\sqrt{10} + \sqrt{15}}{\sqrt{2}}$ | $\frac{2\sqrt{5} + \sqrt{30}}{2}$ |
| (v) $\frac{2 + \sqrt{3}}{3}$ | $\frac{2 + \sqrt{3}}{3}$ |
| (vi) $\frac{\sqrt{2} - 1}{\sqrt{5}}$ | $\frac{\sqrt{10} - \sqrt{5}}{5}$ |

Let me know if you'd like these written in decimal form or further simplified!