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Let's work through the problems shown in your image:

### Problem 1: Simplifying the addition and subtraction of square roots

#### (a) \( 7\sqrt{2} + 8\sqrt{2} \)
- Since both terms have the same square root, we can add the coefficients:
  \[
  (7 + 8)\sqrt{2} = 15\sqrt{2}
  \]

#### (b) \( 5\sqrt{12} + 3\sqrt{75} \)
- Simplify the square roots:
  \[
  \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}, \quad \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}
  \]
- Now substitute the simplified square roots:
  \[
  5\sqrt{12} + 3\sqrt{75} = 5(2\sqrt{3}) + 3(5\sqrt{3}) = 10\sqrt{3} + 15\sqrt{3} = 25\sqrt{3}
  \]

#### (c) \( \frac{7\sqrt{2} + 3\sqrt{18}}{2\sqrt{2}} \)
- Simplify \( \sqrt{18} \):
  \[
  \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}
  \]
- Now substitute back:
  \[
  \frac{7\sqrt{2} + 3(3\sqrt{2})}{2\sqrt{2}} = \frac{7\sqrt{2} + 9\sqrt{2}}{2\sqrt{2}} = \frac{16\sqrt{2}}{2\sqrt{2}} = 8
  \]

#### (d) \( \frac{2\sqrt{5} \times 4\sqrt{10}}{2\sqrt{2}} \)
- Multiply the numerators:
  \[
  2\sqrt{5} \times 4\sqrt{10} = 8\sqrt{50} = 8\sqrt{25 \times 2} = 8 \times 5\sqrt{2} = 40\sqrt{2}
  \]
- Now divide by \( 2\sqrt{2} \):
  \[
  \frac{40\sqrt{2}}{2\sqrt{2}} = \frac{40}{2} = 20
  \]

### Problem 2: Simplifying the multiplication and division of square roots

#### (a) \( \sqrt{4\sqrt{2}} \times \sqrt{3\sqrt{6}} \)
- Simplify:
  \[
  \sqrt{4} = 2, \quad \sqrt{6} = 2\sqrt{

Simplify the given square root expressions, perform addition, subtraction, multiplication, and division of square roots, and solve related problems.

Learn how to simplify square root expressions, rationalize denominators, and calculate the area and perimeter of rectangles with detailed solutions. This guide is perfect for students in Grades 8-10 who are working on algebra and square root operations.

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