Let's work through the problems in the image you provided. Here are detailed solutions for each question:

1. Tentukan hasil dari operasi bilangan berpangkat berikut:

a. $\frac{5^3 \cdot 5^{-2}}{5^2}$

Using the properties of exponents:

$5^3 \cdot 5^{-2} = 5^{3 + (-2)} = 5^1$ $\frac{5^1}{5^2} = 5^{1 - 2} = 5^{-1}$ $5^{-1} = \frac{1}{5}$

So, the answer is $\frac{1}{5}$.

b. $3^5 \cdot 6^{-5}$

Here, both bases are different, and we cannot combine the exponents directly. So the answer is: $3^5 \cdot 6^{-5}$ Which is the final simplified form.

2. Tentukan hasil dari operasi bentuk-bentuk akar berikut:

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, $a = 2\sqrt{3}$ and $b = \sqrt{2}$. So: $(2\sqrt{3})^2 - (\sqrt{2})^2 = 4 \cdot 3 - 2 = 12 - 2 = 10$ Thus, the result is $10$.

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

We can simplify this expression step by step. First, simplify each term inside the parentheses: $\sqrt{5\sqrt{6}} = \sqrt{5} \cdot \sqrt{\sqrt{6}} = \sqrt{5} \cdot (6^{1/4})$ The second term is already in simplest form: $4\sqrt{5}$ Thus, the full expression is: $\sqrt{3} \cdot \left( \sqrt{5} \cdot 6^{1/4} + 4\sqrt{5} \right)$ This expression cannot be simplified further without numerical approximation.

3. Rasionalkan pembagian bentuk akar berikut:

a. $\frac{2\sqrt{10}}{\sqrt{7} - \sqrt{5}}$

To rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator: $\text{Conjugate of } (\sqrt{7} - \sqrt{5}) = (\sqrt{7} + \sqrt{5})$ So: $\frac{2\sqrt{10}}{\sqrt{7} - \sqrt{5}} \cdot \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} + \sqrt{5}} = \frac{2\sqrt{10}(\sqrt{7} + \sqrt{5})}{(\sqrt{7})^2 - (\sqrt{5})^2}$ Simplify the denominator: $(\sqrt{7})^2 - (\sqrt{5})^2 = 7 - 5 = 2$ Thus, the expression becomes: $\frac{2\sqrt{10}(\sqrt{7} + \sqrt{5})}{2} = \sqrt{10}(\sqrt{7} + \sqrt{5})$ So the answer is: $\sqrt{10}(\sqrt{7} + \sqrt{5})$

b. $\frac{\sqrt{5}}{\sqrt{8}}$

Simplify by combining the square roots: $\frac{\sqrt{5}}{\sqrt{8}} = \frac{\sqrt{5}}{2\sqrt{2}} = \frac{\sqrt{10}}{4}$

So, the answer is $\frac{\sqrt{10}}{4}$.

Would you like more details on any part of these solutions? Here are 5 related questions to expand on this topic:

1. How can you simplify expressions involving different bases with exponents?
2. What is the difference of squares formula and where else can it be applied?
3. How do you rationalize denominators involving square roots?
4. What are the steps to simplify nested square root expressions like $\sqrt{5\sqrt{6}}$?
5. How can you handle more complex expressions involving both radicals and exponents?

Tip: When simplifying radicals, always look for perfect squares that can help break down the expression into simpler components.