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It seems you've uploaded an image containing a math problem. Let me analyze it and provide a detailed solution for the question shown in the image.

The image contains two math problems involving exponent rules. I'll analyze and solve both.

### Problem 1:
Given equation:
\[
\frac{x^3}{x^7}
\]
Using the quotient rule for exponents:
\[
\frac{x^a}{x^b} = x^{a-b}
\]
Here, \( a = 3 \) and \( b = 7 \), so:
\[
\frac{x^3}{x^7} = x^{3-7} = x^{-4}
\]
Thus, the simplified expression is:
\[
x^{-4}
\]

### Problem 2:
Given equation:
\[
(4x^5)(-2x^9)
\]
To solve this, we multiply the coefficients and then apply the product rule for exponents:
- Multiply the coefficients: \( 4 \times -2 = -8 \)
- Add the exponents of \( x \): \( 5 + 9 = 14 \)

Thus, the simplified expression is:
\[
-8x^{14}
\]

Would you like further details or any specific clarifications on these problems? 

Here are five additional related questions for further practice:
1. Simplify: \( \frac{y^6}{y^2} \)
2. Simplify: \( (3a^4)(-7a^2) \)
3. Simplify: \( \frac{z^8}{z^3} \times z^{-2} \)
4. Simplify: \( \frac{2x^7}{4x^3} \)
5. Simplify: \( (5x^2y^3)(-3x^4y^2) \)

**Tip:** Always remember that when dividing with the same base, subtract the exponents, and when multiplying with the same base, add the exponents!

Simplify the following expressions: 1. x^3 / x^7, 2. (4x^5)(-2x^9)

This guide simplifies exponential expressions such as x^3 / x^7 and (4x^5)(-2x^9) using the quotient and product rules for exponents. Learn the step-by-step process to master exponent laws, perfect for students in Grades 7-9.

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