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.