Let's start by solving the problems from the image you provided. The questions seem to be about simplifying square roots and performing operations with radicals.

1. Simplifying Expressions:

We will work step by step to simplify the expressions given.

a. $3\sqrt{3} + \sqrt{3}$

Factor out the common term $\sqrt{3}$:

$3\sqrt{3} + \sqrt{3} = (3 + 1)\sqrt{3} = 4\sqrt{3}$

b. $2\sqrt{3} - \sqrt{3}$

Again, factor out the common term $\sqrt{3}$:

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

c. $2\sqrt{5} + 4\sqrt{5}$

Factor out $\sqrt{5}$:

$2\sqrt{5} + 4\sqrt{5} = (2 + 4)\sqrt{5} = 6\sqrt{5}$

d. $\sqrt{8} + \sqrt{2}$

Simplify $\sqrt{8}$ since $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$:

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

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

This combines like terms:

$(1 + 2 - 3)\sqrt{5} = 0\sqrt{5} = 0$

f. $6\sqrt{2} + 4\sqrt{2} - 10\sqrt{2}$

Combine the terms:

$(6 + 4 - 10)\sqrt{2} = 0\sqrt{2} = 0$

g. $2\sqrt{20} - \sqrt{80}$

Simplify both square roots:

$2\sqrt{20} = 2\sqrt{4 \times 5} = 2 \times 2\sqrt{5} = 4\sqrt{5}$ $\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$

Thus, the expression becomes:

$4\sqrt{5} - 4\sqrt{5} = 0$

h. $3\sqrt{32} - 2\sqrt{50}$

Simplify the square roots:

$3\sqrt{32} = 3\sqrt{16 \times 2} = 3 \times 4\sqrt{2} = 12\sqrt{2}$ $2\sqrt{50} = 2\sqrt{25 \times 2} = 2 \times 5\sqrt{2} = 10\sqrt{2}$

Thus, the expression becomes:

$12\sqrt{2} - 10\sqrt{2} = 2\sqrt{2}$

i. $(3\sqrt{2} + 2\sqrt{2})^2$

First, simplify inside the parentheses:

$3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}$

Now square it:

$(5\sqrt{2})^2 = 25 \times 2 = 50$

j. $5\sqrt{3} + 6\sqrt{2} + \sqrt{32} - 3\sqrt{27}$

Simplify $\sqrt{32}$ and $\sqrt{27}$:

$\sqrt{32} = 4\sqrt{2}, \quad \sqrt{27} = 3\sqrt{3}$

Now substitute and simplify:

$5\sqrt{3} + 6\sqrt{2} + 4\sqrt{2} - 3 \times 3\sqrt{3} = 5\sqrt{3} + 6\sqrt{2} + 4\sqrt{2} - 9\sqrt{3}$

Combine like terms:

$(5 - 9)\sqrt{3} + (6 + 4)\sqrt{2} = -4\sqrt{3} + 10\sqrt{2}$

I will continue solving the rest of the expressions. Do you have any questions so far or would you like me to move forward with the remaining ones?