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Let's break down the questions and solve them one by one:

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### 1. Solve the square root expressions:

#### a. \( (2 \sqrt{3} + \sqrt{2})(2 \sqrt{3} - \sqrt{2}) \)

This is a difference of squares:

\[
(a + b)(a - b) = a^2 - b^2
\]

Here:
- \( a = 2 \sqrt{3} \)
- \( b = \sqrt{2} \)

\[
(2 \sqrt{3})^2 - (\sqrt{2})^2 = 4 \times 3 - 2 = 12 - 2 = 10
\]

**Answer:** 10

---

#### b. \( \sqrt{3}(5 \sqrt{6} + 4 \sqrt{5}) \)

Distribute \( \sqrt{3} \):

\[
\sqrt{3} \times 5 \sqrt{6} + \sqrt{3} \times 4 \sqrt{5}
\]

First term: 
\[
\sqrt{3} \times \sqrt{6} = \sqrt{18} = 3 \sqrt{2}, \quad 5 \times 3 \sqrt{2} = 15 \sqrt{2}
\]

Second term:
\[
\sqrt{3} \times \sqrt{5} = \sqrt{15}, \quad 4 \times \sqrt{15} = 4 \sqrt{15}
\]

So, the result is:

\[
15 \sqrt{2} + 4 \sqrt{15}
\]

**Answer:** \( 15 \sqrt{2} + 4 \sqrt{15} \)

---

### 2. Rationalize the following:

#### a. \( \frac{2 \sqrt{10}}{\sqrt{7} - \sqrt{5}} \)

To rationalize this expression, multiply both the numerator and the denominator by the conjugate of the denominator \( \sqrt{7} + \sqrt{5} \):

\[
\frac{2 \sqrt{10}}{\sqrt{7} - \sqrt{5}} \times \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} + \sqrt{5}}
\]

This gives:

\[
\frac{2 \sqrt{10}(\sqrt{7} + \sqrt{5})}{(\sqrt{7})^2 - (\sqrt{5})^2}
\]

First, calculate the denominator:

\[
(\sqrt{7})^2 - (\sqrt{5})^2 = 7 - 5 = 2
\]

Now, the numerator:

\[
2 \sqrt{10} \times \sqrt{7} = 2 \sqrt{70}, \quad 2 \sqrt{10} \times \sqrt{5} = 2 \sqrt{50} = 10 \sqrt{2}
\]

So the full expression becomes:

\[
\frac{2 \sqrt{70} + 10 \sqrt{2}}{2}
\]

This simplifies to:

\[
\sqrt{70} + 5 \sqrt{2}
\]

**Answer:** \( \sqrt{70} + 5 \sqrt{2} \)

---

#### b. \( \frac{\sqrt{5}}{\sqrt{8}} \)

Simplify the square roots:

\[
\frac{\sqrt{5}}{\sqrt{8}} = \frac{\sqrt{5}}{2 \sqrt{2}}
\]

To rationalize, multiply both the numerator and the denominator by \( \sqrt{2} \):

\[
\frac{\sqrt{5}}{2 \sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{10}}{2 \times 2} = \frac{\sqrt{10}}{4}
\]

**Answer:** \( \frac{\sqrt{10}}{4} \)

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Would you like further clarifications or details on any of these steps?

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Here are 5 related questions you can try:

1. Simplify \( (\sqrt{7} + 2)(\sqrt{7} - 2) \).
2. Rationalize \( \frac{1}{\sqrt{5} + 2} \).
3. Multiply \( 3 \sqrt{2} \times 4 \sqrt{3} \).
4. Find the simplified form of \( \frac{\sqrt{12}}{\sqrt{3}} \).
5. Simplify \( 4 \sqrt{6} - 2 \sqrt{6} \).

**Tip:** When rationalizing denominators, always look for the conjugate to simplify the expression efficiently!

Tentukan hasil dari operasi bentuk-bentuk akar berikut: a. (2√3 + √2)(2√3 - √2), b. √3 (5√6 + 4√5), Rasionalkan pembagian bentuk akar berikut: a. 2√10 / (√7 - √5), b. √5 / √8

Learn how to simplify radical expressions and rationalize denominators using conjugates. This exercise involves algebraic operations on square roots and is suitable for students in Grades 9-11.

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