Here are the simplified answers for each problem:

1. $10^6 \cdot 10^5 = 100,000,000,000$
2. $2^4 \cdot 2^3 \cdot 2^2 = 512$
3. $(-4)^8 \cdot (-4)^2 = 1,048,576$
4. $(-y)^8 \cdot (-y)^0 \cdot (-y)^6 = y^{14}$
5. $3^2 \cdot 3 = 27$
6. $\left(\frac{1}{10}\right)^{15} \cdot \left(\frac{1}{10}\right)^{16} = \frac{1}{10^{31}}$
7. $\left(\frac{1}{4}\right) \cdot \left(\frac{1}{4}\right) \cdot \left(\frac{1}{4}\right) \cdot \left(\frac{1}{4}\right) = \frac{1}{256}$
8. $10^{10} \cdot 10^8 \cdot (-2)^7 \cdot (-2)^9 = 65,536,000,000,000,000,000,000$

Do you want details on any specific step?

Here are five related questions:

1. How does the property of exponents apply when multiplying terms with the same base?
2. What happens to the exponents when you multiply powers with the same base?
3. How do you simplify an expression involving negative exponents?
4. What is the significance of the zero exponent in an expression?
5. How do you simplify an expression with a fractional base raised to a power?

Tip: When multiplying terms with the same base, you add the exponents.