Let's solve the square root operations from the image one by one.

1. $5\sqrt{2} + \sqrt{2} - 3\sqrt{2}$

Combine like terms:

$(5 + 1 - 3)\sqrt{2} = 3\sqrt{2}$

So, the result is $\boxed{3\sqrt{2}}$.

2. $3\sqrt{24} - \sqrt{150} + 2\sqrt{54}$

First, simplify the square roots:

• $\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$
• $\sqrt{150} = \sqrt{25 \times 6} = 5\sqrt{6}$
• $\sqrt{54} = \sqrt{9 \times 6} = 3\sqrt{6}$

Now substitute into the expression:

$3\sqrt{24} - \sqrt{150} + 2\sqrt{54} = 3(2\sqrt{6}) - 5\sqrt{6} + 2(3\sqrt{6})$ $= 6\sqrt{6} - 5\sqrt{6} + 6\sqrt{6}$ $= (6 - 5 + 6)\sqrt{6} = 7\sqrt{6}$

So, the result is $\boxed{7\sqrt{6}}$.

3. $2\sqrt{72} - 5\sqrt{32} + 3\sqrt{162}$

First, simplify the square roots:

• $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$
• $\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$
• $\sqrt{162} = \sqrt{81 \times 2} = 9\sqrt{2}$

Now substitute into the expression:

$2\sqrt{72} - 5\sqrt{32} + 3\sqrt{162} = 2(6\sqrt{2}) - 5(4\sqrt{2}) + 3(9\sqrt{2})$ $= 12\sqrt{2} - 20\sqrt{2} + 27\sqrt{2}$ $= (12 - 20 + 27)\sqrt{2} = 19\sqrt{2}$

So, the result is $\boxed{19\sqrt{2}}$.

4. $(4\sqrt{2})(\sqrt{2} + \sqrt{3})$

Distribute $4\sqrt{2}$:

$4\sqrt{2} \times \sqrt{2} + 4\sqrt{2} \times \sqrt{3}$ $= 4(2) + 4\sqrt{6}$ $= 8 + 4\sqrt{6}$

So, the result is $\boxed{8 + 4\sqrt{6}}$.

5. $(3 - \sqrt{5})(3 + \sqrt{5})$

This is a difference of squares:

$(3 - \sqrt{5})(3 + \sqrt{5}) = 3^2 - (\sqrt{5})^2$ $= 9 - 5 = 4$

So, the result is $\boxed{4}$.

Let me know if you need more details or have any questions! Here are some additional questions you can explore:

1. How do you simplify radical expressions like $\sqrt{24}$?
2. Why does $(a - b)(a + b)$ give a difference of squares?
3. Can you expand $(2 + \sqrt{3})(2 - \sqrt{3})$ using the same difference of squares formula?
4. How do you combine like terms with radicals, such as $5\sqrt{6} + 3\sqrt{6}$?
5. How can you simplify the expression $2\sqrt{18} + 3\sqrt{8}$?

Tip:

When simplifying square roots, try factoring numbers into perfect squares for easier simplification.