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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!

Which of the following is equal to (6^14) / ((2^5)(3^7))?

Learn how to simplify the expression (6^14) / ((2^5)(3^7)) using exponent rules and prime factorization. The step-by-step breakdown demonstrates exponent division and prime number properties. This problem is suitable for students in Grades 8-10.

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