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solve the problems. for the first one, assume that any variables in the expression are nonzero. for the second one, only write your answer in scientific notation.

Let's break down and solve both problems.

### Problem 1:  
\[
\frac{(5x)^0 y^{-9}}{(2y)^3 x^0}
\]

#### Steps:
1. **Simplify powers of zero:**
   - Any term raised to the power of zero is 1, provided that the base is non-zero.
   \[
   (5x)^0 = 1 \quad \text{and} \quad x^0 = 1
   \]
   So the expression simplifies to:
   \[
   \frac{1 \cdot y^{-9}}{(2y)^3 \cdot 1}
   \]

2. **Simplify \((2y)^3\):**
   \[
   (2y)^3 = 2^3 \cdot y^3 = 8y^3
   \]
   Now the expression becomes:
   \[
   \frac{y^{-9}}{8y^3}
   \]

3. **Combine the powers of \(y\):**
   Using the property of exponents, \( \frac{y^a}{y^b} = y^{a-b} \):
   \[
   \frac{y^{-9}}{y^3} = y^{-9-3} = y^{-12}
   \]
   So the expression becomes:
   \[
   \frac{1}{8y^{12}}
   \]

Final simplified form:
\[
\boxed{\frac{1}{8y^{12}}}
\]

---

### Problem 2:
\[
(45,000,000)(2,000,000,000)
\]

#### Steps:
1. **Convert each number to scientific notation:**
   - \( 45,000,000 = 4.5 \times 10^7 \)
   - \( 2,000,000,000 = 2 \times 10^9 \)

2. **Multiply the numbers:**
   \[
   (4.5 \times 10^7) \times (2 \times 10^9)
   \]
   Multiply the coefficients \(4.5 \times 2 = 9.0\) and add the exponents of 10:
   \[
   10^7 \times 10^9 = 10^{7+9} = 10^{16}
   \]

Final result:
\[
\boxed{9.0 \times 10^{16}}
\]

---

Do you need any more details or clarifications on these solutions?

Here are 5 related questions to expand on this:
1. How do you simplify negative exponents in a fraction?
2. What happens if the base of an exponent is zero?
3. How would you divide powers with the same base?
4. What is the rule for multiplying numbers in scientific notation?
5. Can scientific notation be applied to very small numbers?

**Tip:** Remember that any term raised to the power of zero is always 1 (as long as the base is nonzero).

Rewrite the expression using only positive exponents, and simplify: (5x)^0 y^-9 / (2y)^3 x^0 and Evaluate the expression (45,000,000)(2,000,000,000) by first writing each number in scientific notation.

Learn how to simplify expressions with exponents, including negative exponents, and solve multiplication problems in scientific notation. This lesson explains key concepts such as exponent rules and scientific notation multiplication with detailed solutions.

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