https://file.aiquickdraw.com/mathbot/file/1727110796460bt0qztv7.jpg

Let's simplify each expression step-by-step:

### a) \( 5\sqrt{20} \)

Simplify \( \sqrt{20} \):
\[
\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}
\]

So, the expression becomes:
\[
5 \times 2\sqrt{5} = 10\sqrt{5}
\]

### b) \( \sqrt{28} + 3\sqrt{7} \)

Simplify \( \sqrt{28} \):
\[
\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}
\]

So, the expression becomes:
\[
2\sqrt{7} + 3\sqrt{7} = 5\sqrt{7}
\]

### c) \( 3\sqrt{8} - 2\sqrt{18} \)

Simplify \( \sqrt{8} \) and \( \sqrt{18} \):
\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \quad \text{and} \quad \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}
\]

So, the expression becomes:
\[
3 \times 2\sqrt{2} - 2 \times 3\sqrt{2} = 6\sqrt{2} - 6\sqrt{2} = 0
\]

### d) \( 2\sqrt{7} \times 3\sqrt{2} \)

Multiply the terms:
\[
(2 \times 3) \times (\sqrt{7} \times \sqrt{2}) = 6\sqrt{14}
\]

### e) \( 2\sqrt{5} \times \sqrt{20} \)

Simplify \( \sqrt{20} \):
\[
\sqrt{20} = 2\sqrt{5}
\]

Multiply the terms:
\[
2\sqrt{5} \times 2\sqrt{5} = 4 \times 5 = 20
\]

### f) \( 15\sqrt{12} \div 5\sqrt{3} \)

Simplify \( \sqrt{12} \):
\[
\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}
\]

Divide the terms:
\[
\frac{15 \times 2\sqrt{3}}{5\sqrt{3}} = \frac{30\sqrt{3}}{5\sqrt{3}} = 6
\]

### g) \( \sqrt{3}(\sqrt{3} + \sqrt{15}) \)

Distribute \( \sqrt{3} \):
\[
\sqrt{3} \times \sqrt{3} + \sqrt{3} \times \sqrt{15} = 3 + \sqrt{45}
\]

Simplify \( \sqrt{45} \):
\[
\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}
\]

So, the expression becomes:
\[
3 + 3\sqrt{5}
\]

### h) \( (\sqrt{7} + \sqrt{2})^2 \)

Expand using the square of a binomial:
\[
(\sqrt{7} + \sqrt{2})^2 = (\sqrt{7})^2 + 2(\sqrt{7})(\sqrt{2}) + (\sqrt{2})^2 = 7 + 2\sqrt{14} + 2 = 9 + 2\sqrt{14}
\]

### i) \( (\sqrt{5} + \sqrt{6})(\sqrt{2} + \sqrt{5}) \)

Expand using FOIL (First, Outside, Inside, Last):
\[
\sqrt{5} \times \sqrt{2} + \sqrt{5} \times \sqrt{5} + \sqrt{6} \times \sqrt{2} + \sqrt{6} \times \sqrt{5} = \sqrt{10} + 5 + \sqrt{12} + \sqrt{30}
\]

Simplify \( \sqrt{12} \):
\[
\sqrt{12} = 2\sqrt{3}
\]

So, the expression becomes:
\[
\sqrt{10} + 5 + 2\sqrt{3} + \sqrt{30}
\]

Would you like more details on any specific problem, or do you have any questions?

### Related Questions:
1. How do you simplify expressions with multiple radicals?
2. What are the rules for multiplying and dividing square roots?
3. How can you simplify expressions with binomials involving radicals?
4. What are common mistakes when working with radicals?
5. How do you expand and simplify expressions using the distributive property?

**Tip:** Always simplify the radicals first before performing other operations; this often makes the calculations easier!

Simplify each expression as presented in the image: a) 5√20 b) √28 + 3√7 c) 3√8 - 2√18 d) 2√7 × 3√2 e) 2√5 × √20 f) 15√12 ÷ 5√3 g) √3(√3 + √15) h) (√7 + √2)^2 i) (√5 + √6)(√2 + √5)

Learn how to simplify radical expressions, including multiplying and adding square roots. This step-by-step guide covers simplifying square roots like √20, expanding binomials with radicals, and applying the distributive property. Suitable for students in Grades 9-11.

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