Solve exponential equations to uncover a hidden message in this engaging math worksheet.

I'll solve each exponential equation step by step to find the hidden message.

Let me work through each problem:

A: 4^x + 7 = 23
- Subtract 7 from both sides: 4^x = 16
- Since 4^2 = 16, then x = 2

B: 8^(x-8) = 8
- Since 8^1 = 8, we have x - 8 = 1
- Therefore x = 9

C: (3^x)^3 = 3
- This simplifies to 3^(3x) = 3^1
- So 3x = 1, which means x = 1/3

D: 5^x = 1/25
- Since 1/25 = 1/5^2 = 5^(-2), then x = -2

E: 7^x = 7
- Since 7^1 = 7, then x = 1

J: (6^x)^2 = 6
- This simplifies to 6^(2x) = 6^1
- So 2x = 1, which means x = 1/2

K: (-3)^x = 729
- Since 729 = 3^6 and (-3)^6 = 729 (even power makes it positive), then x = 6

L: 7^x - 9 = -8
- Add 9 to both sides: 7^x = 1
- Since 7^0 = 1, then x = 0

M: 5^(x-2) = 5
- Since 5^1 = 5, we have x - 2 = 1
- Therefore x = 3

N: x^2 = 7
- Taking square root: x = ±√7
- But looking at the answer choices, this doesn't match any integer. Let me reconsider... Actually x = √7 or x = -√7

Wait, let me check if there's a simpler interpretation. Looking at the pattern, I need to match with the given answers.

S: -4^x = -256
- Divide both sides by -1: 4^x = 256
- Since 4^4 = 256, then x = 4

T: (5^x)^3 = 25
- This simplifies to 5^(3x) = 25 = 5^2
- So 3x = 2, which means x = 2/3

U: 5(2^x) = 5/16
- Divide both sides by 5: 2^x = 1/16
- Since 1/16 = 1/2^4 = 2^(-4), then x = -4

V: 9^(x+x) = 9
- This is 9^(2x) = 9^1
- So 2x = 1, which means x = 1/2

W: 9^x + x = 9
- If x = 1: 9^1 + 1 = 9 + 1 = 10 ≠ 9
- If x = 0: 9^0 + 0 = 1 + 0 = 1 ≠ 9
- Let me try x = 1/2: 9^(1/2) + 1/2 = 3 + 0.5 = 3.5 ≠ 9
- Hmm, this one is tricky. Let me think... Actually if we look for x where 9^x + x = 9, trying x ≈ 0.9 might work but that's not in our list.

Let me reconsider W. Perhaps there's an error in my reading. Let me assume W corresponds to one of the standard answers.

Looking at the answer key provided:
- x = 9 → B
- x = -3 → ?
- x = 7 → ?
- x = -2 → D
- x = 2/3 → T
- x = 8 → ?

Let me recalculate more carefully and match with the given answer options.

From the box at top: x=9, x=-3, x=7, x=-2, x=2/3, x=8

Matching my solutions:
- A: x = 2 (not in list)
- B: x = 9 ✓
- C: x = 1/3 (not in list)
- D: x = -2 ✓
- E: x = 1 (not in list)
- J: x = 1/2 (not in list)
- K: x = 6 (not in list)
- L: x = 0 (not in list)
- M: x = 3 (not in list)
- N: x = √7 (not in list)
- S: x = 4 (not in list)
- T: x = 2/3 ✓
- U: x = -4 (not in list)
- V: x = 1/2 (not in list)
- W: needs solving

Let me re-examine some problems more carefully:

A: 4^x + 7 = 23
4^x = 16 = 4^2, so x = 2

Hmm, none of these match the given answers except B, D, and T. Let me look at this differently - maybe I need to find which letter goes with which answer from the top box.

Actually, looking more carefully at the worksheet structure, I think the student needs to solve each equation and then match the x-value to decode a message using the letters at the top.

Given answers at top: x=9, x=-3, x=7, x=-2, x=2/3, x=8

Let me verify which equations give these specific answers:
- x = 9: B (8^(x-8) = 8 → x-8=1 → x=9) ✓
- x = -2: D (5^x = 1/25 = 5^(-2)) ✓
- x = 2/3: T ((5^x)^3 = 25 → 5^(3x) = 5^2 → 3x=2 → x=2/3) ✓

For the others, I need to find equations that equal x=-3, x=7, or x=8.

Let me check if I misread any:
- Could K be x=-3? (-3)^x = 729. If x=6, (-3)^6 = 729 ✓. Not -3.
- What about other equations?

Actually, I realize I should just provide the solutions I found. The "hidden message" aspect requires matching solved x-values to letters, but without seeing how they map, I'll provide all the x-values I calculated.

Final Answer:
A: x = 2
B: x = 9
C: x = 1/3
D: x = -2
E: x = 1
J: x = 1/2
K: x = 6
L: x = 0
M: x = 3
N: x = √7 or x = -√7
S: x = 4
T: x = 2/3
U: x = -4
V: x = 1/2
W: x ≈ 0.91 (requires numerical solution)