Exponential Equations Worksheet #1 featuring ten problems to solve, each involving exponential expressions with variables in the exponents.

Final Answer:
1. x = 3
2. a = 0
3. p = -1/4
4. x = -2
5. x = 5/3
6. x = 3/7
7. x = -1/3
8. x = 1/3
9. a = 6
10. m = -15/5 → wait, let’s recalculate #10 carefully.

Let me double-check problem #10:

Problem 10: 2^(m+1) = 16^(m+7)

We know 16 = 2^4, so rewrite right side:

→ 2^(m+1) = (2^4)^(m+7)
→ 2^(m+1) = 2^(4(m+7))
→ 2^(m+1) = 2^(4m + 28)

Now set exponents equal:

m + 1 = 4m + 28
Subtract m from both sides:
1 = 3m + 28
Subtract 28:
-27 = 3m
Divide by 3:
m = -9

Ah! I made an error earlier. Correct answer is m = -9.

So correcting all answers with careful checks:

1. 5^(3-2x) = 5^(-x) → 3 - 2x = -x → 3 = x → x = 3 ✔
2. 3^(2a) = 3^(-a) → 2a = -a → 3a = 0 → a = 0 ✔
3. 4^(2p) = 4^(-2p-1) → 2p = -2p -1 → 4p = -1 → p = -1/4 ✔
4. 6^(-2x) = 6^(2-3x) → -2x = 2 - 3x → x = 2? Wait — let's check again.

Wait — problem 4: 6^(-2x) = 6^(2 - 3x)

Set exponents equal:
-2x = 2 - 3x
Add 3x to both sides:
x = 2

But earlier I said x = -2 — that was wrong!

Correcting:

4. x = 2

Let me redo all quickly with care:

1. 3 - 2x = -x → 3 = x → x=3
2. 2a = -a → 3a=0 → a=0
3. 2p = -2p -1 → 4p = -1 → p=-1/4
4. -2x = 2 - 3x → x = 2
5. 8^(x-1) = 2^(x+2); 8=2^3 → (2^3)^(x-1)=2^(x+2) → 2^(3x-3)=2^(x+2) → 3x-3=x+2 → 2x=5 → x=5/2? Wait — original says 8^{x-1} = 2^{x+2}

Yes: 3(x-1) = x+2 → 3x - 3 = x + 2 → 2x = 5 → x = 5/2

Earlier I wrote 5/3 — mistake!

6. 144^(2x+1) = 12^(5x-1); 144=12^2 → (12^2)^(2x+1)=12^(5x-1) → 12^(4x+2)=12^(5x-1) → 4x+2=5x-1 → 2+1=5x-4x → x=3

Wait — earlier I had x=3/7 — big mistake!

Let me fix systematically:

I’ll solve each one properly now:

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1. 5^(3−2x) = 5^(−x)
→ 3 − 2x = −x
→ 3 = x
✔ x = 3

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2. 3^(2a) = 3^(−a)
→ 2a = −a
→ 3a = 0
✔ a = 0

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3. 4^(2p) = 4^(−2p−1)
→ 2p = −2p − 1
→ 4p = −1
✔ p = −1/4

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4. 6^(−2x) = 6^(2−3x)
→ −2x = 2 − 3x
→ x = 2
✔ x = 2

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5. 8^(x−1) = 2^(x+2)
Write 8 as 2³:
(2³)^(x−1) = 2^(x+2)
→ 2^(3x−3) = 2^(x+2)
→ 3x − 3 = x + 2
→ 2x = 5
✔ x = 5/2

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6. 144^(2x+1) = 12^(5x−1)
144 = 12² → (12²)^(2x+1) = 12^(5x−1)
→ 12^(4x+2) = 12^(5x−1)
→ 4x + 2 = 5x − 1
→ 2 + 1 = 5x − 4x
✔ x = 3

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7. 5^(3x+1) = 25^(x+1)
25 = 5² → 5^(3x+1) = (5²)^(x+1) = 5^(2x+2)
→ 3x + 1 = 2x + 2
→ x = 1
Wait — earlier I said x = -1/3 — wrong!

Check:
Left exponent: 3x+1
Right exponent: 2(x+1) = 2x+2
Set equal: 3x+1 = 2x+2 → x = 1
✔ x = 1

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8. 4^(−x+1) = 2^(2x)
4 = 2² → (2²)^(−x+1) = 2^(2x)
→ 2^(−2x + 2) = 2^(2x)
→ −2x + 2 = 2x
→ 2 = 4x
✔ x = 1/2

Earlier I said 1/3 — incorrect.

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9. 64^a = 8^(a+2)
64 = 8² → (8²)^a = 8^(a+2)
→ 8^(2a) = 8^(a+2)
→ 2a = a + 2
✔ a = 2

Earlier I said 6 — wrong.

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10. 2^(m+1) = 16^(m+7)
16 = 2⁴ → 2^(m+1) = (2⁴)^(m+7) = 2^(4m+28)
→ m + 1 = 4m + 28
→ 1 − 28 = 4m − m
→ −27 = 3m
✔ m = −9

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Final corrected answers:

1. x = 3
2. a = 0
3. p = -1/4
4. x = 2
5. x = 5/2
6. x = 3
7. x = 1
8. x = 1/2
9. a = 2
10. m = -9

All verified step-by-step.