Integer Exponents homework worksheet with problems and a riddle to solve.

Let’s solve each problem step by step.

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Problem 1:
4³ × 4⁻⁶ = 4⁵ × 4?

When multiplying powers with the same base, add exponents:
Left side: 3 + (-6) = -3 → 4⁻³
Right side: 5 + ? = -3 → ? = -8

✔ Answer: -8

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Problem 2:
(5⁻³)² = (5⁻⁶)?

Power of a power: multiply exponents → (-3)×2 = -6 → left side is 5⁻⁶
So right side must be (5⁻⁶)¹ → ? = 1

✔ Answer: 1

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Problem 3:
w⁷ / w⁻² = w? / w²

Left side: subtract exponents → 7 - (-2) = 9 → w⁹
Right side: ? - 2 = 9 → ? = 11

Wait — but looking at the answer key, 11 isn’t listed as an option for this problem. Let me recheck.

Actually, let’s write it clearly:

Left: w⁷ ÷ w⁻² = w^(7 - (-2)) = w^9
Right: w^? ÷ w² = w^(? - 2)

Set equal: ? - 2 = 9 → ? = 11

But in the matching table, 11 corresponds to “L” — and that’s used later. But we need to match numbers to letters for the riddle.

Hold on — maybe I made a mistake? Let’s check the original problem again.

It says:
3. w⁷ / w⁻² = w? / w²

Yes, so ? = 11.

But wait — looking at the bottom grid, number 11 is there, mapped to letter L. So that’s fine. We’ll use it when filling the riddle.

✔ Answer: 11

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Problem 4:
(q³)? = q⁻² × q⁸

Right side: -2 + 8 = 6 → q⁶
Left side: (q³)^? = q^(3×?) → 3×? = 6 → ? = 2

✔ Answer: 2

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Problem 5:
(y²)? / y⁶ = y⁻⁷ × y⁷

Right side: -7 + 7 = 0 → y⁰ = 1
Left side: (y²)^? / y⁶ = y^(2×?) / y⁶ = y^(2? - 6)

Set equal: 2? - 6 = 0 → 2? = 6 → ? = 3

✔ Answer: 3

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Problem 6:
2⁻⁸ / 2² = (2⁵)?

Left side: -8 - 2 = -10 → 2⁻¹⁰
Right side: (2⁵)^? = 2^(5×?) → 5×? = -10 → ? = -2

✔ Answer: -2

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Problem 7:
(v⁻²)³ = v⁶ × v?

Left side: -2 × 3 = -6 → v⁻⁶
Right side: 6 + ? = -6 → ? = -12

✔ Answer: -12

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Problem 8:
g³ × g⁻⁴ = g⁶ / g?

Left side: 3 + (-4) = -1 → g⁻¹
Right side: 6 - ? = -1 → ? = 7

✔ Answer: 7

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Problem 9:
8? / 8³ = 1 / 8⁸

Right side: 1/8⁸ = 8⁻⁸
Left side: ? - 3 = -8 → ? = -5

✔ Answer: -5

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Problem 10:
1/r³ = r⁻⁷ × r?

Left side: r⁻³
Right side: -7 + ? = -3 → ? = 4

✔ Answer: 4

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Problem 11:
(10⁻⁵ × 10¹⁴) / 10³ = 10? / 10²

Numerator: -5 + 14 = 9 → 10⁹
Divide by 10³: 9 - 3 = 6 → left side = 10⁶
Right side: ? - 2 = 6 → ? = 8

✔ Answer: 8

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Now, let’s list all answers with their corresponding letters from the grid:

Grid:
| Number | Letter |
|--------|--------|
| 2 | E |
| 6 | D |
| -12 | L |
| -3 | M |
| 1 | O |
| 4 | I |
| -8 | E |
| 9 | A |
| -2 | S |
| -5 | C |
| 7 | S |
| 8 | C |
| 3 | S |
| -7 | U |
| 11 | E |

Now map each problem’s answer to its letter:

1. -8 → E
2. 1 → O
3. 11 → E
4. 2 → E
5. 3 → S
6. -2 → S
7. -12 → L
8. 7 → S
9. -5 → C
10. 4 → I
11. 8 → C

Now fill the riddle blanks using the numbers given under the question:

Riddle: WHAT KINDS OF TRIANGLES ARE THE COLDEST?

Blanks correspond to these positions:

Position: 10 11 4 8 2 5 9 1 7 3 6
Numbers: _ _ _ _ _ _ _ _ _ _ _

We have:

- Position 10 → Problem 10 → 4 → I
- Position 11 → Problem 11 → 8 → C
- Position 4 → Problem 4 → 2 → E
- Position 8 → Problem 8 → 7 → S
- Position 2 → Problem 2 → 1 → O
- Position 5 → Problem 5 → 3 → S
- Position 9 → Problem 9 → -5 → C
- Position 1 → Problem 1 → -8 → E
- Position 7 → Problem 7 → -12 → L
- Position 3 → Problem 3 → 11 → E
- Position 6 → Problem 6 → -2 → S

So the word is:

I C E S O S C E L E S → Wait, that doesn’t make sense.

Wait — let’s write them in order:

Positions: 10, 11, 4, 8, 2, 5, 9, 1, 7, 3, 6
Letters: I, C, E, S, O, S, C, E, L, E, S

That spells: ICESOSC ELES — not meaningful.

Wait — perhaps I misread the position mapping.

Look back at the worksheet:

Under the riddle, it says:

WHAT KINDS OF TRIANGLES ARE THE COLDEST?

Then below:

_ _ _ _ _ _ _ _ _ _ _
10 11 4 8 2 5 9 1 7 3 6

This means:

First blank → Problem 10 → answer 4 → letter I
Second blank → Problem 11 → answer 8 → letter C
Third blank → Problem 4 → answer 2 → letter E
Fourth blank → Problem 8 → answer 7 → letter S
Fifth blank → Problem 2 → answer 1 → letter O
Sixth blank → Problem 5 → answer 3 → letter S
Seventh blank → Problem 9 → answer -5 → letter C
Eighth blank → Problem 1 → answer -8 → letter E
Ninth blank → Problem 7 → answer -12 → letter L
Tenth blank → Problem 3 → answer 11 → letter E
Eleventh blank → Problem 6 → answer -2 → letter S

So the phrase is: I C E S O S C E L E S

That’s “ICESOSCELES” — which is close to “ISOSCELES”!

Ah! Probably a typo or misalignment.

Wait — let’s count the letters: 11 blanks.

“ISOSCELES” has 9 letters. Hmm.

Perhaps it’s “ICE SCALENE”? No.

Wait — maybe I messed up the assignment.

Alternative idea: Maybe the numbers under the blanks refer to the problem numbers, not the answers.

Look again:

The line says:

_ _ _ _ _ _ _ _ _ _ _
10 11 4 8 2 5 9 1 7 3 6

This likely means:

Blank 1 → Problem 10
Blank 2 → Problem 11
Blank 3 → Problem 4
etc.

Which is what I did.

But “ICESO SCELES” doesn’t make sense.

Unless... perhaps the riddle is “WHAT KINDS OF TRIANGLES ARE THE COLDEST?” and the answer is “ISOSCELES” — but we have 11 letters.

Wait — maybe it’s “ICE ISOSCELES” or something.

Another thought: Perhaps the last part is “SCALES” or “SCALE”.

Let’s try to rearrange or see if any letters are wrong.

List of letters in order:

Pos 1 (Prob 10): I
Pos 2 (Prob 11): C
Pos 3 (Prob 4): E
Pos 4 (Prob 8): S
Pos 5 (Prob 2): O
Pos 6 (Prob 5): S
Pos 7 (Prob 9): C
Pos 8 (Prob 1): E
Pos 9 (Prob 7): L
Pos 10 (Prob 3): E
Pos 11 (Prob 6): S

So: I C E S O S C E L E S

If we group: ICE SOS CELES — still not good.

Wait — what if it’s “ISOSCELES” but spelled with extra letters? Or perhaps it’s “ICE” + “ISOSCELES” minus some letters.

Notice: Positions 3 to 11: E S O S C E L E S — that’s almost “ISOSCELES” but starts with E instead of I.

Position 3 is E (from prob 4), but should be I for “ISOSCELES”.

Unless I miscalculated problem 4.

Problem 4: (q³)? = q⁻² × q⁸
Right: q^{6}
Left: q^{3*?} = q^6 → ?=2 → letter E — correct.

Perhaps the riddle is “WHAT KINDS OF TRIANGLES ARE THE COLDEST?” and the punchline is “ISOSCELES” because it sounds like “ice sosceles” — but we have “icesosceles”.

“Icesosceles” can be read as “Ice-sosceles” — meaning cold isosceles triangles.

And “isosceles” is a type of triangle.

So the full answer is “ICESOSCELES” — which is meant to be heard as “Ice Isosceles”.

In many such puzzles, they play on words.

So the final string is: ICESOSCELES

But let's confirm the spelling: I-C-E-S-O-S-C-E-L-E-S — yes.

And the joke is: "What kinds of triangles are the coldest?" -> "Ice-sosceles" (isosceles).

Perfect.

So the completed riddle answer is: ICESOSCELES

But since the instruction is to provide only the final answer for the problems, and the main task was to solve the exponent problems and match to letters, but the user didn't specify what exactly to output.

Looking back at the user's request: "Solve the problem accurately."

The problem is to find the missing exponents and then use them to solve the riddle.

Since the riddle is the ultimate goal, and the final answer expected is probably the solved riddle phrase.

But the response format says: "Final Answer: [Provide only the final correct answer]"

Given that, and since the riddle is the culmination, I think the final answer is the word that completes the riddle.

Moreover, in the context, after solving all, you get "ICESOSCELES", which is the punchline.

So I'll go with that.

But let's double-check one thing: In problem 3, I got 11, which maps to E, and it's used in position 10 of the riddle, giving the second 'E' in "CELES".

All seems consistent.

Final Answer: ICESOSCELES