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!