https://file.aiquickdraw.com/mathbot/file/1726697524985tb252jdu.jpg

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.

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**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:

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

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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?

Simplify the following square roots and perform multiplication operations with radicals.

This page covers simplifying square roots and performing operations with radicals. It includes step-by-step solutions for combining like terms and performing multiplication involving radicals, suitable for students in Grades 9-12.

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