Let's simplify the expression step by step:

$\frac{6a^{5}b^{7}}{-2a^{3}b^{7}}$

Step 1: Simplify the coefficients

First, simplify the numerical coefficients:

$\frac{6}{-2} = -3$

Step 2: Simplify the powers of $a$

We have $a^{5}$ in the numerator and $a^{3}$ in the denominator. When dividing powers of the same base, subtract the exponents:

$\frac{a^{5}}{a^{3}} = a^{5-3} = a^{2}$

Step 3: Simplify the powers of $b$

We have $b^{7}$ in both the numerator and denominator. Since the exponents are the same, they cancel each other out:

$\frac{b^{7}}{b^{7}} = 1$

Final result:

So, the simplified expression is:

$-3a^{2}$

Would you like more details or have any questions about this? Here are some related questions:

1. How do you simplify exponents when multiplying or dividing powers with the same base?
2. What is the general rule for simplifying fractions involving variables?
3. Can negative exponents affect the final result when simplifying fractions?
4. How do you apply the distributive property in algebraic expressions?
5. What happens if there are more than two variables in an expression like this?

Tip: When simplifying powers of variables, always remember to subtract the exponents when dividing and add them when multiplying!