https://file.aiquickdraw.com/mathbot/file/1726090055627cja7ofjc.jpg

Let's go through each of the problems step by step to solve for \( x \).

---

### 6) \( 4^{x+2} = 8^x \)

We start by expressing the bases as powers of 2:
\[
4 = 2^2 \quad \text{and} \quad 8 = 2^3
\]
Rewriting the equation:
\[
(2^2)^{x+2} = (2^3)^x
\]
Simplifying:
\[
2^{2(x+2)} = 2^{3x}
\]
This gives:
\[
2x + 4 = 3x
\]
Solving for \( x \):
\[
4 = x
\]

---

### 7) \( 10^{4x+1} > 100^{x-2} \)

Express \( 100 \) as \( 10^2 \):
\[
10^{4x+1} > (10^2)^{x-2}
\]
Simplifying:
\[
10^{4x+1} > 10^{2(x-2)}
\]
This gives:
\[
10^{4x+1} > 10^{2x-4}
\]
Now, we can equate the exponents:
\[
4x + 1 > 2x - 4
\]
Solving for \( x \):
\[
2x > -5
\]
\[
x > -\frac{5}{2}
\]

---

### 8) \( 36^{2x+4} = 216^{x+5} \)

Express the bases in terms of powers of 6:
\[
36 = 6^2 \quad \text{and} \quad 216 = 6^3
\]
Rewriting the equation:
\[
(6^2)^{2x+4} = (6^3)^{x+5}
\]
Simplifying:
\[
6^{4x+8} = 6^{3(x+5)}
\]
Equating the exponents:
\[
4x + 8 = 3x + 15
\]
Solving for \( x \):
\[
x = 7
\]

---

### 9) \( \frac{1}{81} < 9^{2x-4} \)

Express 81 and 9 as powers of 3:
\[
81 = 3^4 \quad \text{and} \quad 9 = 3^2
\]
Rewriting the equation:
\[
3^{-4} < (3^2)^{2x-4}
\]
Simplifying:
\[
3^{-4} < 3^{4x-8}
\]
Equating the exponents:
\[
-4 < 4x - 8
\]
Solving for \( x \):
\[
4x > 4
\]
\[
x > 1
\]

---

### 10) \( 32^{3x-4} > 128^{4x+3} \)

Express 32 and 128 as powers of 2:
\[
32 = 2^5 \quad \text{and} \quad 128 = 2^7
\]
Rewriting the equation:
\[
(2^5)^{3x-4} > (2^7)^{4x+3}
\]
Simplifying:
\[
2^{5(3x-4)} > 2^{7(4x+3)}
\]
Equating the exponents:
\[
15x - 20 > 28x + 21
\]
Solving for \( x \):
\[
-13x > 41
\]
\[
x < -\frac{41}{13}
\]

---

### Final Answers:
1. \( x = 4 \)
2. \( x > -\frac{5}{2} \)
3. \( x = 7 \)
4. \( x > 1 \)
5. \( x < -\frac{41}{13} \)

Let me know if you'd like further details on any of these steps!

---

Here are 5 additional questions for further exploration:
1. Can you solve exponential equations using logarithms?
2. How do you handle inequalities when solving exponential equations?
3. What happens if the bases of the exponential equations are not powers of the same number?
4. Can you solve exponential equations with negative exponents?
5. How would you graph exponential inequalities?

**Tip**: Always look for common bases in exponential equations to simplify solving!

Solve for x in the given exponential equations: 4^(x+2) = 8^x, 10^(4x+1) > 100^(x-2), 36^(2x+4) = 216^(x+5), 1/81 < 9^(2x-4), 32^(3x-4) > 128^(4x+3).

Learn how to solve complex exponential equations like 4^(x+2) = 8^x and inequalities such as 10^(4x+1) > 100^(x-2) using exponent rules and logarithmic properties. Explore detailed solutions suitable for high school students.

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