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Let's solve the given problems step by step.

### Problem 1

Given:
\[ a * b = (a + b) \times (a - b) \]
We need to find:
\[ 27 * 9 \]

Using the given operation:
\[ 27 * 9 = (27 + 9) \times (27 - 9) \]
\[ = 36 \times 18 \]
\[ = 648 \]

### Problem 2

Given:
\[ a * b = a^2 + 2b \]
We need to find:
\[ 10 * 6 \text{ and } 5 * (2 * 8) \]

First, calculate \( 10 * 6 \):
\[ 10 * 6 = 10^2 + 2 \times 6 \]
\[ = 100 + 12 \]
\[ = 112 \]

Now, calculate \( 5 * (2 * 8) \):
\[ 2 * 8 = 2^2 + 2 \times 8 \]
\[ = 4 + 16 \]
\[ = 20 \]

Then,
\[ 5 * 20 = 5^2 + 2 \times 20 \]
\[ = 25 + 40 \]
\[ = 65 \]

### Problem 3

Given:
\[ a * b = \frac{3a - b}{2} \]
We need to find:
\[ (25 * 12) * (10 * 5) \]

First, calculate \( 25 * 12 \):
\[ 25 * 12 = \frac{3 \times 25 - 12}{2} \]
\[ = \frac{75 - 12}{2} \]
\[ = \frac{63}{2} \]
\[ = 31.5 \]

Now, calculate \( 10 * 5 \):
\[ 10 * 5 = \frac{3 \times 10 - 5}{2} \]
\[ = \frac{30 - 5}{2} \]
\[ = \frac{25}{2} \]
\[ = 12.5 \]

Finally, calculate \( 31.5 * 12.5 \):
\[ = 31.5 \times 12.5 \]
\[ = 393.75 \]

Thus, the solutions are:

1. \( 27 * 9 = 648 \)
2. \( 10 * 6 = 112 \) and \( 5 * (2 * 8) = 65 \)
3. \( (25 * 12) * (10 * 5) = 393.75 \)

Do you have any questions or need further details?

Here are five more questions you can ask:
1. How do you solve quadratic equations?
2. What are the properties of logarithms?
3. Can you explain the fundamental theorem of calculus?
4. How do you find the derivative of a function?
5. What are the applications of matrices in real life?

**Tip:** When dealing with complex expressions, break them down into smaller, manageable parts to simplify the calculations.

Learn how to solve mathematical problems involving integers and fractions step by step. This problem covers key algebraic concepts, including the distributive property and operations with fractions. Suitable for students in Grades 7-9.

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