https://file.aiquickdraw.com/mathbot/file/1726836260921mhwlzpnq.jpg

solve the problems. (assume that any variables in the expression are nonzero)

Let's solve the two problems shown in the image.

---

### Problem 9:
The expression is:

\[
\frac{(x^6)^0}{x^{-7}}
\]

Step-by-step solution:
1. **Simplify the numerator**: Any number or expression raised to the power of 0 is equal to 1:
   \[
   (x^6)^0 = 1
   \]

2. The expression now becomes:
   \[
   \frac{1}{x^{-7}}
   \]

3. **Rewrite using positive exponents**: To remove the negative exponent, use the property \( x^{-a} = \frac{1}{x^a} \):
   \[
   \frac{1}{x^{-7}} = x^7
   \]

So, the simplified expression is:

\[
x^7
\]

---

### Problem 10:
The expression is:

\[
\frac{(2n)^{-3}}{(2n^3)^0}
\]

Step-by-step solution:
1. **Simplify the denominator**: As stated before, any expression raised to the power of 0 is equal to 1:
   \[
   (2n^3)^0 = 1
   \]

2. The expression now becomes:
   \[
   \frac{(2n)^{-3}}{1} = (2n)^{-3}
   \]

3. **Rewrite using positive exponents**:
   \[
   (2n)^{-3} = \frac{1}{(2n)^3}
   \]

4. Expand the cube:
   \[
   (2n)^3 = 2^3 \cdot n^3 = 8n^3
   \]

So, the simplified expression is:

\[
\frac{1}{8n^3}
\]

---

Do you have any questions or want further details on these steps?

### Related Questions:
1. How do you simplify expressions with multiple negative exponents?
2. What is the general rule for simplifying expressions raised to the power of zero?
3. How do you handle variables in exponents when multiplying and dividing terms?
4. What are the steps to expand powers of binomials, like \((a + b)^n\)?
5. How does changing the base affect exponent rules?

### Tip:
Always simplify powers of zero first in any expression; it makes the problem much easier to handle!

Rewrite the expressions using only positive exponents, and simplify. (Assume that any variables in the expression are nonzero)

Learn how to simplify algebraic expressions involving zero and negative exponents. This detailed step-by-step solution demonstrates how to apply the rules of exponents to solve problems like (x^6)^0 / x^{-7} and (2n)^{-3} / (2n^3)^0. Suitable for high school students in Grades 9-11.

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