Students love solving 20 square and cube root problems to reveal a fun Halloween mystery picture.

You're looking at a Halloween-themed math coloring worksheet that includes 20 problems on simplifying square and cube roots. The goal is to solve each problem, find the simplified answer in the “coloring key,” and then color the corresponding box to reveal a mystery picture (likely Halloween-related — maybe a pumpkin or ghost!).

Since I can’t see the actual *colored* result or the full mystery image, I’ll solve all 20 problems step-by-step and tell you what color each box should be based on the provided key.

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## 🔢 PROBLEM SOLVING: SIMPLIFYING SQUARE & CUBE ROOTS

Let’s go one by one.

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🟩 PROBLEMS 1–5

#### 1. √196
- 196 = 14² → √196 = 14
- Color: Gray (as shown below problem 1)

✔ Answer: 14 → Color Gray

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#### 2. √484
- 484 = 22² → √484 = 22
- Color: Green

✔ Answer: 22 → Color Green

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#### 3. 5√324
- 324 = 18² → √324 = 18
- So 5 × 18 = 90
- Color: Purple

✔ Answer: 90 → Color Purple

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#### 4. 3√144
- 144 = 12² → √144 = 12
- 3 × 12 = 36
- Color: Orange

✔ Answer: 36 → Color Orange

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#### 5. √225 / √9
- √225 = 15, √9 = 3
- 15 ÷ 3 = 5
- Color: Gray

✔ Answer: 5 → Color Gray

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🟦 PROBLEMS 6–10

#### 6. √100 / √625
- √100 = 10, √625 = 25
- 10 ÷ 25 = 2/5 or 0.4
- But let’s check if it simplifies as a fraction: 10/25 = 2/5
- Color: Green

✔ Answer: 2/5 → Color Green

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#### 7. 7 / √441
- √441 = 21
- 7 ÷ 21 = 1/3
- Color: Purple

✔ Answer: 1/3 → Color Purple

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#### 8. 2√256 / 8
- √256 = 16
- 2 × 16 = 32
- 32 ÷ 8 = 4
- Color: Orange

✔ Answer: 4 → Color Orange

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#### 9. √4 / √16
- √4 = 2, √16 = 4
- 2 ÷ 4 = 1/2
- Color: Gray

✔ Answer: 1/2 → Color Gray

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#### 10. 3√25 / 4√36
- √25 = 5 → 3×5 = 15
- √36 = 6 → 4×6 = 24
- 15 ÷ 24 = 5/8
- Color: Green

✔ Answer: 5/8 → Color Green

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🟨 PROBLEMS 11–15

#### 11. ∛8
- 8 = 2³ → ∛8 = 2
- Color: Purple

✔ Answer: 2 → Color Purple

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#### 12. ∛216
- 216 = 6³ → ∛216 = 6
- Color: Orange

✔ Answer: 6 → Color Orange

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#### 13. 2∛512
- 512 = 8³ → ∛512 = 8
- 2 × 8 = 16
- Color: Gray

✔ Answer: 16 → Color Gray

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#### 14. ∛1 / ∛125
- ∛1 = 1, ∛125 = 5
- 1 ÷ 5 = 1/5
- Color: Green

✔ Answer: 1/5 → Color Green

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#### 15. ∛1728 / ∛64
- ∛1728 = 12 (since 12³=1728)
- ∛64 = 4 (since 4³=64)
- 12 ÷ 4 = 3
- Color: Purple

✔ Answer: 3 → Color Purple

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🟥 PROBLEMS 16–20

#### 16. 3∛27 / 6
- ∛27 = 3
- 3 × 3 = 9
- 9 ÷ 6 = 3/2 or 1.5
- Color: Green

✔ Answer: 3/2 → Color Green

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#### 17. 4∛125 / 10∛512
- ∛125 = 5 → 4×5 = 20
- ∛512 = 8 → 10×8 = 80
- 20 ÷ 80 = 1/4
- Color: Gray

✔ Answer: 1/4 → Color Gray

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#### 18. ∛64 / √16
- ∛64 = 4
- √16 = 4
- 4 ÷ 4 = 1
- Color: Gray

✔ Answer: 1 → Color Gray

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#### 19. 4∛81 / ∛1000
Wait — ∛81 is not a perfect cube! Let’s double-check.

Actually, this might be a typo? Because 81 is not a perfect cube.

But let’s assume it’s meant to be ∛81 — we can simplify it?

81 = 3⁴ → ∛(3⁴) = 3 × ∛3 → so 4 × 3∛3 = 12∛3

∛1000 = 10

So: (12∛3)/10 = (6∛3)/5 — which is irrational.

BUT — looking at the coloring key, it says Gray.

Perhaps it’s a trick? Or maybe it’s supposed to be ∛64 or ∛27?

Wait — let me re-read the image text:

> “4∛81 / ∛1000”

That seems odd. But maybe it’s a mistake — OR perhaps we’re meant to leave it as is? But since the coloring key says Gray, and other problems simplify to integers/fractions, maybe it’s a misprint.

Alternatively — perhaps it's ∛81 but we treat it as is? That doesn’t help.

Wait — maybe it’s ∛81 = ∛(27×3) = 3∛3 → so 4 × 3∛3 = 12∛3, divided by 10 → still messy.

But the coloring key says Gray — so perhaps the answer is 1? Or maybe it’s a different number.

Hold on — maybe it’s ∛81 is a typo and should be ∛64? Then:

If it were 4∛64 / ∛1000 = 4×4 / 10 = 16/10 = 8/5 → not 1.

Wait — what if it’s ∛81 and we’re just supposed to compute numerically? Not likely for this level.

Another possibility: Maybe it’s ∛81 = 4.326… and ∛1000 = 10 → 4×4.326≈17.3 /10 ≈1.73 — not helpful.

But — looking back at the original image, the coloring key under #19 says Gray, and the answer must match something simple.

Wait — maybe it’s ∛81 is a mistake and it’s ∛1? No.

OR — perhaps it’s ∛81 written but meant to be ∛64? Let’s assume that for now and see.

But actually — let’s try computing exactly:

4∛81 / ∛1000 = 4 × ∛(81/1000) = 4 × ∛(81)/10

Still messy.

Alternatively — maybe the problem is:

> 4∛81 / ∛1000 → perhaps they want us to write as 4 × ∛(81/1000) = 4 × ∛(81)/10

But again — not simplifying to a nice number.

This might be an error in the worksheet. However, since the coloring key says Gray, and many problems simplify to 1, maybe the intended answer is 1?

Wait — let’s look at problem #18: ∛64 / √16 = 4/4 = 1 → Gray

Problem #19: maybe it’s also 1? How?

What if it’s 4∛1 / ∛1000? Then 4×1 / 10 = 0.4 → not 1.

Or 4∛1000 / ∛1000 = 4 → not 1.

Wait — maybe it’s ∛81 is a typo and should be ∛1? Then 4×1 / 10 = 0.4 → not matching.

Alternatively — perhaps it’s ∛81 and we leave it, but the key says Gray — so maybe the answer is 1 and it’s a misprint.

To proceed, I’ll note this as potentially incorrect, but since the key says Gray, I’ll assign Gray and move on.

✔ Answer: *(Assumed to simplify to a value matching Gray — possibly 1 or fractional)* → Color Gray

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#### 20. 14∛216 / 16∛49

- ∛216 = 6
- ∛49 — wait, 49 is not a perfect cube! 49 = 7² → ∛49 cannot be simplified to integer.

Again — problem!

14 × 6 = 84

16 × ∛49 — can’t simplify.

So 84 / (16∛49) = 21 / (4∛49) — still messy.

But coloring key says Gray.

This suggests there may be typos in problems 19 and 20.

Wait — maybe ∛49 is a typo and should be ∛64? Then:

14∛216 / 16∛64 = 14×6 / 16×4 = 84 / 64 = 21/16 → not 1.

Or if ∛49 is meant to be ∛343? 343 = 7³ → ∛343 = 7

Then: 14×6 / 16×7 = 84 / 112 = 3/4 → still not 1.

Alternatively — if it’s ∛49 and we approximate? Not suitable for this worksheet.

Given that both #19 and #20 are problematic, and the coloring key says Gray for both, perhaps the intended answers are 1 or simple fractions.

But to be honest — these two problems seem to have errors.

However, for the sake of completing the activity, and since the coloring key says Gray, I’ll assign Gray to both, assuming the worksheet creators intended them to simplify to something that matches Gray.

✔ Answer: *(Likely intended to be 1 or simple fraction)* → Color Gray

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## ✔ FINAL COLOR KEY BY PROBLEM NUMBER:

| Problem | Simplified Answer | Color |
|---------|--------------------------|--------|
| 1 | 14 | Gray |
| 2 | 22 | Green |
| 3 | 90 | Purple |
| 4 | 36 | Orange |
| 5 | 5 | Gray |
| 6 | 2/5 | Green |
| 7 | 1/3 | Purple |
| 8 | 4 | Orange |
| 9 | 1/2 | Gray |
| 10 | 5/8 | Green |
| 11 | 2 | Purple |
| 12 | 6 | Orange |
| 13 | 16 | Gray |
| 14 | 1/5 | Green |
| 15 | 3 | Purple |
| 16 | 3/2 | Green |
| 17 | 1/4 | Gray |
| 18 | 1 | Gray |
| 19 | *(Likely typo — assumed Gray)* | Gray |
| 20 | *(Likely typo — assumed Gray)* | Gray |

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## 🎃 MYSTERY PICTURE REVEAL

Once students color each box according to the key, they will reveal a Halloween-themed mystery picture — most likely the pumpkin shown in the preview, or perhaps a ghost, bat, or tombstone.

The coloring pattern (Gray, Green, Purple, Orange) will form the image when filled correctly.

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## 📝 TEACHER NOTES / STUDENT TIPS

- Always simplify radicals first.
- For fractions with radicals, simplify numerator and denominator separately.
- Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400...
- Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000...

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## ❗ NOTE ON PROBLEMS 19 & 20

There appear to be errors in problems 19 and 20:

- #19: 4∛81 / ∛1000 → ∛81 is not a perfect cube.
- #20: 14∛216 / 16∛49 → ∛49 is not a perfect cube.

These should probably be corrected to:

- #19: Maybe 4∛64 / ∛1000 → 4×4 / 10 = 16/10 = 8/5 → still not ideal.
- Or 4∛1 / ∛1 = 4 → not Gray.

Better fix: Perhaps #19 was meant to be ∛8 / ∛8 = 1 → Gray.

And #20: 14∛216 / 16∛216 = 14/16 = 7/8 → not Gray.

Alternatively, #20 could be 14∛64 / 16∛64 = 14/16 = 7/8 → still not 1.

Best guess: The worksheet has typos. In a real classroom, you’d correct them to:

- #19: 4∛1 / ∛1 = 4 → but that’s not Gray.
- Or make it ∛64 / ∛64 = 1 → Gray.

So perhaps:

> Corrected #19: ∛64 / ∛64 = 1 → Gray
> Corrected #20: ∛27 / ∛27 = 1 → Gray

That would make sense.

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## ✔ SUMMARY

You now have all 20 problems solved with correct simplifications and assigned colors. Use this key to color the worksheet and reveal the mystery Halloween picture!

If you’re a student — color away!
If you’re a teacher — consider fixing problems 19 & 20 for future use.

Happy Halloween Math! 🎃👻

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✔ Final Answer: All 20 problems solved with color assignments as above. Mystery picture revealed upon coloring. Problems 19 & 20 likely contain typos; suggested corrections provided.