Let's solve the problems using the properties of exponents, specifically focusing on the product of powers property which states that $a^m \times a^n = a^{m+n}$.

1. $10^6 \cdot 10^5$
   Using the product of powers property:
   $10^{6+5} = 10^{11}$

2. $2^4 \cdot 2^3 \cdot 2^2$
   Again, using the product of powers property:
   $2^{4+3+2} = 2^9$

3. $(-4)^8 \cdot (-4)^2$
   Here, the bases are the same, so add the exponents:
   $(-4)^{8+2} = (-4)^{10}$

4. $(-y)^8 \cdot (-y)^0 \cdot (-y)^6$
   The exponent for $(-y)^0$ is 0, and any base raised to the power of 0 equals 1. Thus:
   $(-y)^{8+0+6} = (-y)^{14}$

5. $3^2 \cdot 3$
   Remember that $3$ is the same as $3^1$. Adding exponents gives:
   $3^{2+1} = 3^3$

6. $\left(\frac{1}{10}\right)^{15} \cdot \left(\frac{1}{10}\right)^{16}$
   Add the exponents since the bases are the same:
   $\left(\frac{1}{10}\right)^{15+16} = \left(\frac{1}{10}\right)^{31}$

7. $\left(\frac{1}{a}\right) \cdot \left(\frac{1}{a}\right) \cdot \left(\frac{1}{a}\right) \cdot \left(\frac{1}{a}\right)$
   This can be written as $\left(\frac{1}{a}\right)^4$.

8. $10^{10} \cdot 10^8 \cdot (-2)^7 \cdot (-2)^9$
   For the powers of 10: $10^{10+8} = 10^{18}$ For the powers of $-2$: $(-2)^{7+9} = (-2)^{16}$ Thus, the final expression is $10^{18} \cdot (-2)^{16}$.

Would you like more details on any of the solutions, or do you have any other questions?

Here are some related questions to explore:

1. How do you simplify expressions with different bases?
2. What happens when you multiply numbers with different signs?
3. How do you deal with fractional exponents?
4. What is the importance of the zero exponent rule?
5. Can you apply the power of a power property to the above problems?

Tip: When multiplying powers with the same base, always add the exponents, regardless of whether the base is positive or negative.