Simplify the Complex Fraction: Dividing Powers and Managing Negative Exponents

Let's systematically simplify both expressions and then compare them:

Simplifying the First Expression:

$\frac{7^2 \cdot 7^{-8}}{7^3 \cdot (-7)^4}$

• Apply Product of Powers Rule to the numerator: $7^2 \cdot 7^{-8} = 7^{2 + (-8)} = 7^{-6}$.

• Use Power of a Power Rule in the denominator for $(-7)^4 = (-1)^4 \cdot 7^4 = 7^4$ because $(-1)^4 = 1$.

• Simplify the denominator: $7^3 \cdot 7^4 = 7^{3+4} = 7^7$.

• Apply Quotient of Powers Rule: $\frac{7^{-6}}{7^7} = 7^{-6-7} = 7^{-13}$.

Simplifying the Second Expression:

$\frac{7^2 \cdot 7^{-9}}{7^3 \cdot (-7)^4}$

• Apply Product of Powers Rule to the numerator: $7^2 \cdot 7^{-9} = 7^{2 + (-9)} = 7^{-7}$.

• The denominator is the same as before: $7^3 \cdot 7^4 = 7^7$.

• Apply Quotient of Powers Rule: $\frac{7^{-7}}{7^7} = 7^{-7-7} = 7^{-14}$.

Comparison:

• The first expression simplifies to $7^{-13}$.

• The second expression simplifies to $7^{-14}$.

• Since $-13 > -14$,
  $7^{-13} > 7^{-14}$.

Therefore, the first expression is greater than the second expression. The correct choice is: $>$.