Worksheet for practicing exponent properties with simplification exercises.

Let’s go through each problem step by step using the properties of exponents.

We’ll use these key rules:
- When multiplying powers with the same base: add exponents → a^m × a^n = a^(m+n)
- When dividing powers with the same base: subtract exponents → a^m ÷ a^n = a^(m−n)
- When raising a power to another power: multiply exponents → (a^m)^n = a^(m×n)
- Any nonzero number to the power 0 is 1 → a^0 = 1
- Negative exponents mean reciprocal → a^(-n) = 1/a^n, but we want positive exponents only.

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Top Section:

1. 4⁶ · 4⁸
→ Same base, multiply → add exponents: 6 + 8 = 14
→ Answer: 4¹⁴

2. 3¹¹
→ Already simplified.
→ Answer: 3¹¹

3. 6¹⁶ / 6⁷
→ Same base, divide → subtract exponents: 16 − 7 = 9
→ Answer: 6⁹

4. (5⁵)⁴
→ Power raised to power → multiply exponents: 5 × 4 = 20
→ Answer: 5²⁰

5. 2⁷⁴
→ Already simplified.
→ Answer: 2⁷⁴

6. 12⁸ / 12⁶
→ Subtract exponents: 8 − 6 = 2
→ Answer: 12²

7. 8¹ · 8⁶
→ Add exponents: 1 + 6 = 7
→ Answer: 8⁷

8. (17⁶)⁴
→ Multiply exponents: 6 × 4 = 24
→ Answer: 17²⁴

9. (13³)⁵
→ Multiply exponents: 3 × 5 = 15
→ Answer: 13¹⁵

10. 3¹ · 3⁸
→ Add exponents: 1 + 8 = 9
→ Answer: 3⁹

11. 14¹⁰
→ Already simplified.
→ Answer: 14¹⁰

12. 9¹¹ / 9⁷
→ Subtract exponents: 11 − 7 = 4
→ Answer: 9⁴

13. 7⁶ · 7⁹
→ Add exponents: 6 + 9 = 15
→ Answer: 7¹⁵

14. (4⁷)⁵
→ Multiply exponents: 7 × 5 = 35
→ Answer: 4³⁵

15. 10²¹ / 10¹⁶
→ Subtract exponents: 21 − 16 = 5
→ Answer: 10⁵

16. 48¹⁴
→ Already simplified.
→ Answer: 48¹⁴

17. 2⁹ / 2⁴
→ Subtract exponents: 9 − 4 = 5
→ Answer: 2⁵

18. 18⁻¹⁰
→ Negative exponent → write as 1 over positive exponent, but instruction says “single term with positive exponent” — so we leave it as 18⁻¹⁰? Wait — no! The instruction says “write the answer as a single term with a positive exponent.” So we must convert negative to positive by moving to denominator? But that would be a fraction. Hmm.

Wait — let’s re-read: “Write the answer as a single term with a positive exponent.” That means if you have a negative exponent, you must rewrite it so the exponent is positive — which usually means writing it as 1 over the base to the positive power. But that’s not a “single term” in the sense of just base^exponent — unless they allow fractions? Actually, looking at context, maybe they expect us to leave it as is? No — the rule is clear: positive exponent only.

Actually, 18⁻¹⁰ = 1 / 18¹⁰ — but that’s two terms (numerator and denominator). The instruction says “single term”. So perhaps they mean don’t leave negative exponents — so we must express it as 18⁻¹⁰ is invalid, so we write it as 1/18¹⁰ — but again, not a single term.

Wait — maybe I misread. Let me check the original problem again.

Looking back: “Simplify each expression using the properties of exponents. Write the answer as a single term with a positive exponent.”

So for 18⁻¹⁰, since it’s already a single term but has a negative exponent, we must rewrite it to have a positive exponent — which requires writing it as a fraction. But “single term” might still allow fractions? Or perhaps they consider 18⁻¹⁰ acceptable? No — because it says “positive exponent”.

Actually, in many textbooks, when they say “write with positive exponents”, they accept 1/a^n as the form. And “single term” might mean not expanded or anything. So I think for 18⁻¹⁰, the answer is 1/18¹⁰ — but let’s see if there’s another way.

Alternatively, maybe the problem expects us to leave it as 18⁻¹⁰? But that violates “positive exponent”.

I think the safest is to follow the rule: convert negative exponents to positive by writing as reciprocal.

But let’s look at other problems — for example, 5⁻⁴ · 5⁻⁴ — that will also give negative exponent.

Perhaps for now, I’ll note that 18⁻¹⁰ should be written as 1/18¹⁰ — but let’s confirm with the challenge section later.

Actually, let’s proceed and come back.

For now, I’ll write:

18. 18⁻¹⁰ → To have positive exponent, it’s 1/18¹⁰ — but is that allowed? The instruction says “single term” — a fraction is one term? In algebra, yes, 1/18¹⁰ is a single term.

But let’s see what the expected answer is. Perhaps they want us to leave it as 18⁻¹⁰? No, because it says “positive exponent”.

I think I made a mistake — let me check the image again mentally.

Actually, in the image, problem 18 is "18⁻¹⁰" — and it's in the grid. Looking at similar problems, like 5⁻⁴ · 5⁻⁴, which will be 5⁻⁸, then converted to 1/5⁸.

So I think for consistency, we should convert all negative exponents to positive by writing as fractions.

But the instruction says "single term with a positive exponent" — so for 18⁻¹⁰, the answer is \frac{1}{18^{10}} — but that might not be considered a "single term" in some contexts. However, in math, a rational expression like that is often accepted.

To be safe, let's assume that for negative exponents, we write them as 1 over the base to the positive power.

So:

18. 18⁻¹⁰ → 1/18¹⁰

But let's continue and see.

19. (8¹¹)⁴
→ Multiply exponents: 11 × 4 = 44
→ Answer: 8⁴⁴

20. 5⁻⁴ · 5⁻⁴
→ Add exponents: -4 + (-4) = -8
→ Then convert to positive: 1/5⁸
→ Answer: 1/5⁸

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Challenge Section:

21. 9⁷ / 9⁴
→ Subtract exponents: 7 − 4 = 3
→ Answer: 9³

22. 25² / 25⁻³
→ Subtract exponents: 2 − (-3) = 2 + 3 = 5
→ Answer: 25⁵

23. (7³)⁷
→ Multiply exponents: 3 × 7 = 21
→ Answer: 7²¹

24. (4⁵)⁶
→ Multiply exponents: 5 × 6 = 30
→ Answer: 4³⁰

25. 5⁻¹ · (5⁴)²
First, simplify (5⁴)² = 5⁸
Then, 5⁻¹ · 5⁸ = 5⁻¹⁺⁸ = 5⁷
→ Answer: 5⁷

26. 6⁻⁴ · 6⁻²
→ Add exponents: -4 + (-2) = -6
→ Convert to positive: 1/6⁶
→ Answer: 1/6⁶

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Now, let's list all answers clearly.

But wait — for problem 18 and 20 and 26, we have fractions. Is that acceptable? The instruction says "single term with a positive exponent". A fraction like 1/18¹⁰ can be written as 18⁻¹⁰, but that has negative exponent. So to have positive exponent, we must write it as a fraction.

In many educational contexts, this is acceptable. For example, Khan Academy and others accept 1/x^n as the form with positive exponent.

So I'll go with that.

Final Answers:

Top Section:

1. 4¹⁴
2. 3¹¹
3. 6⁹
4. 5²⁰
5. 2⁷⁴
6. 12²
7. 8⁷
8. 17²⁴
9. 13¹⁵
10. 3⁹
11. 14¹⁰
12. 9⁴
13. 7¹⁵
14. 4³⁵
15. 10⁵
16. 48¹⁴
17. 2⁵
18. 1/18¹⁰
19. 8⁴⁴
20. 1/5⁸

Challenge:

21. 9³
22. 25⁵
23. 7²¹
24. 4³⁰
25. 5⁷
26. 1/6⁶

But let me double-check problem 22: 25² / 25⁻³ = 25^(2 - (-3)) = 25^5 — correct.

Problem 25: 5⁻¹ · (5⁴)² = 5⁻¹ · 5⁸ = 5⁷ — correct.

Problem 26: 6⁻⁴ · 6⁻² = 6⁻⁶ = 1/6⁶ — correct.

All seem good.

Now, for the final answer, since the user didn't specify which part, but the image has multiple problems, I think we need to provide all answers.

But the response format says "Final Answer: [Provide only the final correct answer]" — but there are many answers.

Perhaps I should list them in order.

Since it's a worksheet, maybe list them as per the grid.

But to make it clear, I'll list them numbered as I did.

However, the instruction says "ONLY THEN provide the final answer" — implying one answer, but here there are many.

Perhaps for this task, since it's multiple problems, the final answer is the set of all simplified expressions.

I think I should present them in a boxed format as a list.

But let's see the response format: it says "Final Answer: [text]" — so I can write all answers there.

To be precise, I'll write them in order.

Also, for problem 18, 20, 26, I used fractions. Is there a way to write them without fractions? Only if we allow negative exponents, but the instruction forbids that.

Another thought: in some contexts, "single term" might mean not a fraction, so perhaps for negative exponents, they expect us to leave them as is? But that contradicts "positive exponent".

I recall that in the properties of exponents, when simplifying, if you get a negative exponent, you rewrite it with a positive exponent by taking reciprocal.

So I think it's correct.

Perhaps the problem 18 is 18^{-10}, and they want 1/18^{10}, which is fine.

So I'll proceed.

Final Answer:

1. \( 4^{14} \)
2. \( 3^{11} \)
3. \( 6^{9} \)
4. \( 5^{20} \)
5. \( 2^{74} \)
6. \( 12^{2} \)
7. \( 8^{7} \)
8. \( 17^{24} \)
9. \( 13^{15} \)
10. \( 3^{9} \)
11. \( 14^{10} \)
12. \( 9^{4} \)
13. \( 7^{15} \)
14. \( 4^{35} \)
15. \( 10^{5} \)
16. \( 48^{14} \)
17. \( 2^{5} \)
18. \( \frac{1}{18^{10}} \)
19. \( 8^{44} \)
20. \( \frac{1}{5^{8}} \)
21. \( 9^{3} \)
22. \( 25^{5} \)
23. \( 7^{21} \)
24. \( 4^{30} \)
25. \( 5^{7} \)
26. \( \frac{1}{6^{6}} \)

But to make it neat, perhaps write without LaTeX in the final answer, but since it's text, I can use superscripts.

In the response, I can write:

Final Answer:
4¹⁴, 3¹¹, 6⁹, 5²⁰, 2⁷⁴, 12², 8⁷, 17²⁴, 13¹⁵, 3⁹, 14¹⁰, 9⁴, 7¹⁵, 4³⁵, 10⁵, 48¹⁴, 2⁵, 1/18¹⁰, 8⁴⁴, 1/5⁸, 9³, 25⁵, 7²¹, 4³⁰, 5⁷, 1/6⁶

But for clarity, since it's a worksheet, perhaps list them in rows as in the image.

The image has 4 columns and 6 rows for top, and 3 columns for challenge.

Top section: 6 rows x 4 columns = 24 problems? Let's count.

In the image description, I listed 20 for top? No, I think I miscounted.

Let me recount the problems from the image description.

From the user's image description:

Top section has:

Row 1: 4⁶·4⁸, 3¹¹, 6¹⁶/6⁷, (5⁵)⁴ → 4 problems

Row 2: 2⁷⁴, 12⁸/12⁶, 8¹·8⁶, (17⁶)⁴ → 4 problems

Row 3: (13³)⁵, 3¹·3⁸, 14¹⁰, 9¹¹/9⁷ → 4 problems

Row 4: 7⁶·7⁹, (4⁷)⁵, 10²¹/10¹⁶, 48¹⁴ → 4 problems

Row 5: 2⁹/2⁴, 18⁻¹⁰, (8¹¹)⁴, 5⁻⁴·5⁻⁴ → 4 problems

That's 5 rows x 4 = 20 problems for top.

Then challenge section:

Row 1: 9⁷/9⁴, 25²/25⁻³, (7³)⁷ → 3 problems

Row 2: (4⁵)⁶, 5⁻¹·(5⁴)², 6⁻⁴·6⁻² → 3 problems

Total 26 problems.

In my earlier list, I have 26 answers.

For the final answer, I'll list them in order as per the grid.

So:

First row: 4¹⁴, 3¹¹, 6⁹, 5²⁰

Second row: 2⁷⁴, 12², 8⁷, 17²⁴

Third row: 13¹⁵, 3⁹, 14¹⁰, 9⁴

Fourth row: 7¹⁵, 4³⁵, 10⁵, 48¹⁴

Fifth row: 2⁵, 1/18¹⁰, 8⁴⁴, 1/5⁸

Challenge first row: 9³, 25⁵, 7²¹

Challenge second row: 4³⁰, 5⁷, 1/6⁶

Now, for the fractions, I think it's correct.

So Final Answer will be this list.

To box it, I'll write it as a sequence.

Since the response format might expect a single string, but I think it's fine.

I'll write: