https://file.aiquickdraw.com/mathbot/file/1727445866899bj9gznen.jpg

Pls no more explanation, proceed to solution only thanks. 

Here are the solutions to the exponential equations from the image:

1. \( 64 = 8^{-x-2} \)

Rewrite both sides as powers of 2:
\[
64 = 2^6, \quad 8 = 2^3
\]
\[
2^6 = (2^3)^{-x-2}
\]
\[
2^6 = 2^{-3(x+2)}
\]
Equating exponents:
\[
6 = -3(x+2)
\]
Solve for \( x \):
\[
6 = -3x - 6
\]
\[
12 = -3x
\]
\[
x = -4
\]

---

2. \( 3^{2x} = 81 \)

Rewrite 81 as a power of 3:
\[
81 = 3^4
\]
\[
3^{2x} = 3^4
\]
Equating exponents:
\[
2x = 4
\]
\[
x = 2
\]

---

3. \( 16^{-x} = \frac{1}{64} \)

Rewrite both sides as powers of 2:
\[
16 = 2^4, \quad 64 = 2^6
\]
\[
(2^4)^{-x} = 2^{-6}
\]
\[
2^{-4x} = 2^{-6}
\]
Equating exponents:
\[
-4x = -6
\]
\[
x = \frac{3}{2}
\]

---

4. \( 3^{4x} = 9^{x+1} \)

Rewrite 9 as a power of 3:
\[
9 = 3^2
\]
\[
3^{4x} = (3^2)^{x+1}
\]
\[
3^{4x} = 3^{2(x+1)}
\]
Equating exponents:
\[
4x = 2(x+1)
\]
\[
4x = 2x + 2
\]
\[
2x = 2
\]
\[
x = 1
\]

---

5. \( 4^{x+1} = 64 \)

Rewrite 64 as a power of 4:
\[
64 = 4^3
\]
\[
4^{x+1} = 4^3
\]
Equating exponents:
\[
x + 1 = 3
\]
\[
x = 2
\]

---

Let me know if you'd like further details or have any questions.

Here are 5 related questions to consider:

1. How would you solve an equation like \( 5^{2x} = 25^x \)?
2. Can you solve \( 7^{3x+1} = 49^{x+2} \)?
3. What method do you use when the bases in an exponential equation are different?
4. How would you solve \( 2^{x+3} = 32 \)?
5. Can you find the value of \( x \) if \( 9^{x} = 27 \)?

**Tip**: Always try to express both sides of an exponential equation using the same base for easier comparison!

Solve for the value of x in the given exponential equations.

Step-by-step solutions for solving exponential equations such as 64 = 8^(-x-2), 3^(2x) = 81, and more. This guide covers algebraic methods and exponent rules, suitable for students in Grades 10-12.

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