Let's go through each of the problems step by step.

a. $\sqrt{2} \cdot \sqrt{3}$

Using the property of square roots: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ $\sqrt{2} \cdot \sqrt{3} = \sqrt{2 \cdot 3} = \sqrt{6}$

b. $\sqrt{5} \cdot \sqrt{5}$

This is simplified using: $\sqrt{a} \cdot \sqrt{a} = a$ $\sqrt{5} \cdot \sqrt{5} = 5$

c. $\sqrt{6} \cdot \sqrt{5}$

Applying the same square root multiplication rule: $\sqrt{6} \cdot \sqrt{5} = \sqrt{6 \cdot 5} = \sqrt{30}$

d. $2 \sqrt{2} \cdot 4 \sqrt{3}$

First, multiply the numerical coefficients $2 \cdot 4 = 8$, then the square roots: $\sqrt{2} \cdot \sqrt{3} = \sqrt{2 \cdot 3} = \sqrt{6}$ So, the full expression becomes: $2 \sqrt{2} \cdot 4 \sqrt{3} = 8 \sqrt{6}$

e. $5^3 \cdot \sqrt{6} \cdot 6^3 \cdot \sqrt{5}$

First, simplify the powers and products: $5^3 = 125, \quad 6^3 = 216$ Now multiply the square roots: $\sqrt{6} \cdot \sqrt{5} = \sqrt{6 \cdot 5} = \sqrt{30}$ So, the full expression becomes: $125 \cdot 216 \cdot \sqrt{30}$ We can calculate: $125 \cdot 216 = 27000$ Thus, the result is: $27000 \sqrt{30}$

f. $\frac{\sqrt[3]{10}}{\sqrt[3]{5}}$

This can be simplified as: [ \frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}} ] [ \frac{\sqrt[3]{10}}{\sqrt[3]{5}} = \sqrt[3]{\frac{10}{5}} = \sqrt[3]{2} ]

g. $\frac{\sqrt{24}}{\sqrt{6}}$

Using the property of square roots: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ $\frac{\sqrt{24}}{\sqrt{6}} = \sqrt{\frac{24}{6}} = \sqrt{4} = 2$

h. $\frac{10 \sqrt{6}}{5 \sqrt{2}}$

First, simplify the numerical coefficients: $\frac{10}{5} = 2$ Now, simplify the square roots: $\frac{\sqrt{6}}{\sqrt{2}} = \sqrt{\frac{6}{2}} = \sqrt{3}$ So, the full expression becomes: $2 \cdot \sqrt{3} = 2\sqrt{3}$

Let me know if you need any additional details!

Here are 5 related questions:

1. How do you simplify expressions involving cube roots like in problem (f)?
2. Can you apply the same square root rules for fractional powers, such as $\sqrt{x^2}$?
3. What is the general rule for multiplying or dividing radical expressions?
4. How do you deal with exponents in radical expressions, such as $5^3$?
5. What are some strategies for simplifying nested radicals like $\sqrt{a} \cdot \sqrt{b}$?

Tip: Always look for opportunities to simplify both numerical coefficients and radicals separately to make problems easier to solve!