Exponent Puzzle Worksheet for 6th - 8th Grade | Lesson Planet - Free Printable

It looks like you've uploaded an image of a "Exponent Puzzle" — a crossword-style puzzle where each clue involves simplifying expressions with exponents, and the answers go into the grid. However, I can't view images directly, so I can't see the actual clues or grid.

But based on your description and the visible text in your message (like "E) APPLE: 4² × 4³ × 4⁵ = 512"), I can help solve this type of puzzle if you provide:

- The list of clues (ACROSS and DOWN)
- Or describe what’s in the image

But since I can see part of the puzzle already, let's work with what we have.

---

Step-by-step Analysis of Clue E:

> E) APPLE: 4² × 4³ × 4⁵ = 512

Wait — that seems incorrect.

Let’s check the math:

We are multiplying powers of 4:
$$
4^2 \times 4^3 \times 4^5
$$

Using exponent rules:
$$
4^{2+3+5} = 4^{10}
$$

Now calculate $ 4^{10} $:
- $ 4^2 = 16 $
- $ 4^4 = (4^2)^2 = 16^2 = 256 $
- $ 4^8 = (4^4)^2 = 256^2 = 65,536 $
- $ 4^{10} = 4^8 \times 4^2 = 65,536 \times 16 = 1,048,576 $

So:
$$
4^2 \times 4^3 \times 4^5 = 4^{10} = 1,048,576
$$

But the puzzle says it equals 512, which is not correct.

Wait — maybe the expression is different?

Let’s check: Is it possible they meant 2 instead of 4?

Try:
$$
2^2 \times 2^3 \times 2^5 = 2^{2+3+5} = 2^{10} = 1024
$$
Still not 512.

But:
$$
2^9 = 512
$$

Hmm.

Wait — perhaps the clue is:
$$
2^2 \times 2^3 \times 2^4 = 2^{9} = 512
$$

Yes! That works.

So maybe the original clue has a typo, or perhaps it's written as $ 2^2 \times 2^3 \times 2^4 $, but was misread.

Alternatively, maybe it's:
$$
8^2 \times 8^1 = 64 \times 8 = 512
$$
Or even:
$$
(2^3)^2 = 8^2 = 64
$$

But 512 is $ 2^9 $, or $ 8^3 $, or $ 4^4.5 $? Not nice.

Wait — $ 512 = 2^9 = 8^3 = 4^{4.5} $? No, not helpful.

But $ 512 = 2^9 $, and $ 2^9 = (2^3)^3 = 8^3 $

So maybe the clue is $ 8^3 = 512 $

But in the image, it says:
> E) APPLE: 4² × 4³ × 4⁵ = 512

That’s wrong. Because:
- $ 4^2 = 16 $
- $ 4^3 = 64 $
- $ 4^5 = 1024 $
- $ 16 × 64 = 1024 $
- $ 1024 × 1024 = 1,048,576 $

So clearly, 4² × 4³ × 4⁵ ≠ 512

Therefore, either:
- There’s a typo in the clue
- Or the puzzle is asking to solve for the missing exponent?

Wait — maybe the clue is:
> $ 4^x \times 4^y \times 4^z = 512 $

And you need to find the values?

But 512 is not a power of 4.

Check:
- $ 4^1 = 4 $
- $ 4^2 = 16 $
- $ 4^3 = 64 $
- $ 4^4 = 256 $
- $ 4^5 = 1024 $

So 512 is between $ 4^4 $ and $ 4^5 $. Not a power of 4.

But $ 512 = 2^9 $, and $ 4^x = (2^2)^x = 2^{2x} $, so $ 4^x = 512 = 2^9 $ → $ 2x = 9 $ → $ x = 4.5 $, not integer.

So 512 cannot be written as a single power of 4, nor as a product of integer powers of 4 unless the result is larger.

So the equation:
> $ 4^2 \times 4^3 \times 4^5 = 512 $

is false.

Thus, likely there’s a mistake in the puzzle or in the transcription.

---

Let's assume the puzzle wants us to simplify exponent expressions and fill in the grid.

Here’s how to solve such puzzles:

#### General Rules:
1. $ a^m \times a^n = a^{m+n} $
2. $ a^m \div a^n = a^{m-n} $
3. $ (a^m)^n = a^{m \times n} $
4. $ a^0 = 1 $
5. $ a^{-n} = 1/a^n $

---

Example Clues (common in such puzzles):

Suppose the clues are:

Across:
1. $ 2^3 $
2. $ 3^2 $
3. $ 5^2 $
4. $ 2^5 $
5. $ 10^2 $

Then answers:
1. 8
2. 9
3. 25
4. 32
5. 100

Down:
1. $ 2^4 $
2. $ 3^3 $
3. $ 2^6 $
4. $ 5^3 $

Answers:
1. 16
2. 27
3. 64
4. 125

Then you fill the grid accordingly.

---

But since I can’t see the full puzzle, I can’t fill the grid.

However, if you provide:

- The list of clues (ACROSS and DOWN)
- Or the image content (type out the clues)

I can solve it completely.

---

But wait — you said “E) APPLE: 4² × 4³ × 4⁵ = 512”

Maybe “APPLE” is the answer? Like, the word "APPLE" is hidden in the grid?

But that doesn’t make sense unless the puzzle uses letters.

Wait — perhaps the puzzle is not numerical? Maybe it’s a word puzzle where the answers are words related to exponents?

But the instruction says:
> "Solve each problem... write each expression"

And then:
> "Use a calculator if you want"

So it’s likely numerical.

---

Conclusion:

There is likely a typo in clue E.

Because:
- $ 4^2 \times 4^3 \times 4^5 = 4^{10} = 1,048,576 $
- But 512 = $ 2^9 $, $ 8^3 $, $ 4^{4.5} $ — not valid for integer exponents.

So unless the base is 2, not 4, it doesn’t work.

Let’s suppose the clue was:
> $ 2^2 \times 2^3 \times 2^4 = 2^{9} = 512 $

Then it would be correct.

So perhaps the puzzle meant:
> $ 2^2 \times 2^3 \times 2^4 = 512 $

Then the answer is 512, and you write “512” in the grid.

But the clue says “4²”, not “2²”.

So either:
- It’s a typo
- Or it’s a trick question
- Or the puzzle wants you to evaluate the expression regardless

But the instruction says: “show what the exponent to use” — maybe it’s asking for the exponent?

Wait — perhaps the clue is:
> $ 4^x \times 4^3 \times 4^5 = 512 $

Then solve for x?

Let’s try that.

$ 4^x \times 4^3 \times 4^5 = 4^{x+3+5} = 4^{x+8} = 512 $

But $ 512 = 2^9 $, $ 4^{x+8} = (2^2)^{x+8} = 2^{2(x+8)} $

Set equal:
$$
2^{2(x+8)} = 2^9 \Rightarrow 2(x+8) = 9 \Rightarrow x+8 = 4.5 \Rightarrow x = -3.5
$$

Not nice.

So unlikely.

---

Final Guess:

The clue might be:
> $ 2^2 \times 2^3 \times 2^4 = 512 $

Which is true, because:
- $ 2^2 = 4 $
- $ 2^3 = 8 $
- $ 2^4 = 16 $
- $ 4 × 8 = 32 $
- $ 32 × 16 = 512 $

Yes!

So likely the base is 2, not 4.

So if the clue is:
> $ 2^2 \times 2^3 \times 2^4 = 512 $

Then the answer is 512, and you write it in the grid.

But in your image, it says 4², which is wrong.

---

Request:

Please type out the clues from the puzzle (both ACROSS and DOWN), or describe the image.

Then I can solve the entire puzzle step by step.

Alternatively, if you confirm that the clue is indeed:
> $ 4^2 \times 4^3 \times 4^5 = ? $

Then the correct answer is:
> $ 4^{10} = 1,048,576 $

And you should write 1048576 in the grid.

But since it says "= 512", that’s incorrect.

---

Summary:

✔ Correct value:
$ 4^2 \times 4^3 \times 4^5 = 4^{10} = 1,048,576 $

✘ Claimed value: 512 → False

So either:
- There's a typo in the puzzle
- Or the base is 2, not 4

If the base were 2, then:
$ 2^2 \times 2^3 \times 2^4 = 2^9 = 512 $ → ✔ Correct

So likely, the puzzle meant 2, not 4.

---

Final Answer:

The expression $ 4^2 \times 4^3 \times 4^5 $ equals $ 1,048,576 $, not 512. Therefore, the given equation is incorrect. If the intended expression was $ 2^2 \times 2^3 \times 2^4 $, then it correctly evaluates to 512.

To proceed, please provide the full list of clues from the puzzle, and I’ll solve it completely.