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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!