Examples of dividing exponents with different bases, illustrating mathematical expressions for educational purposes.

Let’s solve each of these four problems one by one. The key idea is: when bases are different, we try to rewrite them with the same base if possible, then use exponent rules. If they can’t be rewritten with the same base, we leave them as is or simplify numerically if needed.

---

Problem 1:
\[
\frac{13^2}{5^4}
\]

- 13 and 5 are both prime numbers — no common base.
- So we cannot combine exponents.
- We can compute numerator and denominator separately:

Numerator: \(13^2 = 169\)
Denominator: \(5^4 = 5 \times 5 \times 5 \times 5 = 625\)

So:
\[
\frac{169}{625}
\]

This fraction doesn’t reduce (since 169 = 13² and 625 = 5⁴ — no common factors).

✔ Final for Problem 1: \(\frac{169}{625}\)

---

Problem 2:
\[
\frac{9^4}{3^6}
\]

Note: 9 = 3² → so rewrite 9⁴ as (3²)⁴ = 3^(2×4) = 3⁸

Now:
\[
\frac{3^8}{3^6} = 3^{8-6} = 3^2 = 9
\]

✔ Final for Problem 2: \(9\)

---

Problem 3:
\[
\frac{25^3}{125^1}
\]

Note: 25 = 5², 125 = 5³

Rewrite:

Numerator: \(25^3 = (5^2)^3 = 5^{6}\)
Denominator: \(125^1 = 5^3\)

So:
\[
\frac{5^6}{5^3} = 5^{6-3} = 5^3 = 125
\]

✔ Final for Problem 3: \(125\)

---

Problem 4:
\[
\frac{64^{-4}}{4^{-2}}
\]

First, note: 64 = 4³? Let’s check: 4³ = 64 → YES!

So rewrite 64⁻⁴ as (4³)⁻⁴ = 4^(3 × -4) = 4⁻¹²

Now expression becomes:
\[
\frac{4^{-12}}{4^{-2}} = 4^{-12 - (-2)} = 4^{-10}
\]

We can also write this as:
\[
\frac{1}{4^{10}}
\]

But let’s see if we want to compute it numerically? Probably not necessary unless asked. But since other answers were simplified, let’s keep it as \(4^{-10}\) or \(\frac{1}{4^{10}}\). However, maybe we should express in simplest form.

Alternatively, note that 4¹⁰ = (2²)¹⁰ = 2²⁰, but again, probably overkill.

Since the problem says “divide”, and we’ve simplified using exponent rules, \(4^{-10}\) is acceptable. But often teachers prefer positive exponents.

So:
\[
\frac{1}{4^{10}} = \frac{1}{1048576}
\]

Wait — let me verify 4¹⁰:

4¹ = 4
4² = 16
4³ = 64
4⁴ = 256
4⁵ = 1024
4⁶ = 4096
4⁷ = 16384
4⁸ = 65536
4⁹ = 262144
4¹⁰ = 1048576 ✔

So yes, \(\frac{1}{1048576}\)

But perhaps leaving as \(4^{-10}\) is fine? Let’s see what the context is. Since others were computed to integers or simple fractions, maybe compute it.

Actually, let’s double-check our exponent rule:

\[
\frac{a^m}{a^n} = a^{m-n}
\]

Here: m = -12, n = -2 → m - n = -12 - (-2) = -10 → correct.

So either \(4^{-10}\) or \(\frac{1}{4^{10}}\) or \(\frac{1}{1048576}\)

I think for consistency with previous problems (which gave numerical values), we’ll go with the fractional form.

✔ Final for Problem 4: \(\frac{1}{1048576}\)

---

Wait — let me recheck Problem 4 because negative exponents might trip us up.

Original: \(\frac{64^{-4}}{4^{-2}}\)

Another way: move terms with negative exponents to opposite side of fraction.

Recall: \(a^{-n} = \frac{1}{a^n}\), so:

\[
\frac{64^{-4}}{4^{-2}} = \frac{1/64^4}{1/4^2} = \frac{4^2}{64^4}
\]

Now, 64 = 4³ → 64⁴ = (4³)⁴ = 4¹²

So:
\[
\frac{4^2}{4^{12}} = 4^{2-12} = 4^{-10} = \frac{1}{4^{10}} = \frac{1}{1048576}
\]

Same result. Good.

---

Now, compiling all final answers:

Problem 1: \(\frac{169}{625}\)
Problem 2: \(9\)
Problem 3: \(125\)
Problem 4: \(\frac{1}{1048576}\)

But wait — the user didn’t specify whether to list all or just one? Looking back at the image description, there are four separate expressions. So likely, we need to provide all four answers.

However, the instruction says: “Solve the problem accurately.” — and the image shows four problems under one title. So I believe we are to solve all four.

But in the response format, it says “Final Answer: [Provide only the final correct answer]” — singular. Hmm.

Looking again: the user uploaded an image containing a task — which appears to be four separate division problems. So probably, the expected output is to solve all four.

But the format says “final answer” — maybe we should present them as a set?

Perhaps the system expects each answered individually? But the instruction says “the problem” — singular.

Wait — looking at the original request: “You are an educational assistant helping a student solve homework problems.” and “uploaded an image containing a task.”

The image has four expressions labeled implicitly as separate items. In many such contexts, especially in worksheets, each is a separate problem.

Given that, and since the assistant is to “solve the problem accurately”, and there are four, I think we must provide all four.

But the format says “Final Answer: [only the final correct answer]” — implying one box.

Perhaps we should list them clearly.

Maybe the intended interpretation is that these are examples under one topic, and we’re to demonstrate how to handle each — but still, the answer should include all results.

To be safe, I’ll present all four answers in order, separated clearly.

Alternatively, perhaps the user wants each solved, and since ExplainAnswer is false, we skip explanation and just give final answers.

But the critical rule says: “Complete ALL reasoning and explanations BEFORE writing the Final Answer.” — which I did above.

Then in Final Answer, put the results.

I think best to write:

Final Answer:
For \(\frac{13^2}{5^4}\): \(\frac{169}{625}\)
For \(\frac{9^4}{3^6}\): \(9\)
For \(\frac{25^3}{125^1}\): \(125\)
For \(\frac{64^{-4}}{4^{-2}}\): \(\frac{1}{1048576}\)

But the format says “only the final correct answer” — maybe they expect a single value? That doesn't make sense.

Another thought: perhaps the image is meant to show types of problems, and the actual task is implied? But no, the user said “solve the problem” referring to the image content.

Given ambiguity, but based on standard practice, I'll assume we need to output all four solutions.

Perhaps format as a list.

Let me check the initial instructions again: “If you're solving math or logic problems, verify your answer before writing it in the Final Answer section.”

And “Final Answer: [Provide only the final correct answer]”

In multi-part problems, sometimes “final answer” means all parts.

I think safest is to present them clearly.

Since ExplainAnswer is false, I won’t explain in the output — but I already did the reasoning above.

So in Final Answer, I’ll write:

\[
\boxed{\dfrac{169}{625}},\quad \boxed{9},\quad \boxed{125},\quad \boxed{\dfrac{1}{1048576}}
\]

But the format might not support multiple boxes. Alternatively, state them in text.

Given the constraints, I’ll write:

Final Answer:
\(\frac{169}{625}\), \(9\), \(125\), \(\frac{1}{1048576}\)

Yes.

But let me confirm once more calculations:

Problem 1: 13²=169, 5⁴=625 → 169/625 ✔️
Problem 2: 9⁴=(3²)⁴=3, /3⁶=3²=9 ✔️
Problem 3: 25³=(5²)³=5⁶, 125=5³, so 5⁶/5³=5³=125 ✔️
Problem 4: 64⁻⁴ / 4⁻² = (4³)⁻⁴ / 4⁻² = 4⁻¹² / 4⁻² = 4⁻¹⁰ = 1/4¹⁰ = 1/1048576 ✔️

All correct.

──────────────────────────────────────

Final Answer:
\(\frac{169}{625}\), \(9\), \(125\), \(\frac{1}{1048576}\)