Let's go through the problems on the worksheet step by step:

1. $46^0$

• Any non-zero number raised to the power of zero equals 1.
• Answer: 1

2. $w^0$, where $w \neq 0$

• As in the previous problem, any non-zero number raised to the power of zero equals 1.
• Answer: 1

3. $8^{-3}$

• Negative exponents mean reciprocal, so $8^{-3} = \frac{1}{8^3} = \frac{1}{512}$.
• Answer: $\frac{1}{512}$

4. $y^{-9}$

• Again, negative exponents indicate the reciprocal: $y^{-9} = \frac{1}{y^9}$.
• Answer: $\frac{1}{y^9}$

5. $\frac{1}{c^6}$

• Since $\frac{1}{c^6}$ is already a fraction, you express it as $c^{-6}$.
• Answer: $c^{-6}$

6. $\frac{1}{10^5}$

• Similarly, $\frac{1}{10^5}$ can be written as $10^{-5}$.
• Answer: $10^{-5}$

7. $9^4 \cdot 9^{-6}$

• Use the property $a^m \cdot a^n = a^{m+n}$, so $9^4 \cdot 9^{-6} = 9^{4 + (-6)} = 9^{-2} = \frac{1}{9^2} = \frac{1}{81}$.
• Answer: $\frac{1}{81}$

8. $y^{-9} \cdot y^3$

• Similarly, $y^{-9} \cdot y^3 = y^{-9+3} = y^{-6} = \frac{1}{y^6}$.
• Answer: $\frac{1}{y^6}$

9. $\frac{x^{-8}}{x^{-12}}$

• Use the property $\frac{a^m}{a^n} = a^{m-n}$, so $\frac{x^{-8}}{x^{-12}} = x^{-8 - (-12)} = x^{4}$.
• Answer: $x^4$

10. $\frac{d^{-13}}{d^{-2}}$

• Similar to the previous one: $\frac{d^{-13}}{d^{-2}} = d^{-13 - (-2)} = d^{-11} = \frac{1}{d^{11}}$.
• Answer: $\frac{1}{d^{11}}$

11. Simplify $8^{-7} \cdot 8^7 \cdot 10^4 \cdot 10^{-4}$

• First, simplify the powers of 8: $8^{-7} \cdot 8^7 = 8^{0} = 1$.
• Then, simplify the powers of 10: $10^4 \cdot 10^{-4} = 10^{4+(-4)} = 10^{0} = 1$.
• The final result is $1 \cdot 1 = 1$.
• Answer: 1

12. Multiselect - Select all of the expressions that are simplified:

• $n^4$ - Yes, this is simplified.

• $\frac{1}{n^{-5}}$ - No, this can be written as $n^5$.

• $n^6 \cdot n^{-8}$ - No, this simplifies to $n^{-2} = \frac{1}{n^2}$.

• $n^7 \cdot p^8$ - Yes, this is simplified.

• $\frac{1}{n^3}$ - Yes, this is simplified.

• Answer: $n^4$, $n^7 \cdot p^8$, and $\frac{1}{n^3}$.

Would you like more details on any of these problems? Here are five related questions:

1. What is the rule for multiplying numbers with the same base but different exponents?
2. How do you simplify expressions with negative exponents?
3. How can you express $a^{-n}$ as a fraction?
4. What happens when you raise a number to the power of zero?
5. How do you simplify the division of exponents with the same base?

Tip: Always simplify exponents by using the rules of addition or subtraction depending on whether the terms are multiplied or divided.