The given expression is:

$\frac{6^{14}}{(2^5)(3^7)}$

We can start by breaking down $6^{14}$ in terms of its prime factors:

$6 = 2 \times 3$ Thus, $6^{14}$ can be written as:

$6^{14} = (2 \times 3)^{14} = 2^{14} \times 3^{14}$

Now, substitute this into the original expression:

$\frac{6^{14}}{(2^5)(3^7)} = \frac{2^{14} \times 3^{14}}{2^5 \times 3^7}$

We can simplify this by dividing the powers of 2 and 3 separately:

$\frac{2^{14}}{2^5} = 2^{14 - 5} = 2^9$ $\frac{3^{14}}{3^7} = 3^{14 - 7} = 3^7$

Thus, the simplified expression is:

$2^9 \times 3^7$

Now, let's compare this with the given answer choices. The correct answer is:

$(2^9)(3^7)$

This corresponds to the fourth option.

Would you like a deeper explanation or have any more questions?

Here are 5 related questions to explore this concept further:

1. How would you simplify $\frac{6^{10}}{(2^3)(3^5)}$?
2. Can you express $12^{6}$ in terms of its prime factors?
3. What is the result of $\frac{8^5}{2^4}$?
4. How would you simplify $\frac{9^{8}}{3^{6}}$?
5. What happens if we multiply $2^9 \times 3^7$ by $6^2$?

Tip: Remember that when dividing powers with the same base, you subtract the exponents!