The OCR did not extract the equations correctly. I'll proceed by solving each equation manually based on the visual content from the uploaded image.

Given problems:

1. $64 = 8^{x - 2}$
2. $3^{2x} = 81$
3. $16^{-x} = \frac{1}{64}$
4. $3^{4x} = 9^{x + 1}$
5. $4^{x + 1} = 64$

Let's solve each one step-by-step.

1) $64 = 8^{x - 2}$

Express 64 and 8 as powers of 2: $64 = 2^6 \quad \text{and} \quad 8 = 2^3$ Thus, the equation becomes: $2^6 = (2^3)^{x - 2} \implies 2^6 = 2^{3(x - 2)}$ Equating exponents: $6 = 3(x - 2) \implies 6 = 3x - 6 \implies 3x = 12 \implies x = 4$

2) $3^{2x} = 81$

Express 81 as a power of 3: $81 = 3^4$ So, $3^{2x} = 3^4 \implies 2x = 4 \implies x = 2$

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

Express 16 and 64 as powers of 2: $16 = 2^4 \quad \text{and} \quad 64 = 2^6$ Therefore, $(2^4)^{-x} = \frac{1}{2^6} \implies 2^{-4x} = 2^{-6}$ Equating exponents: $-4x = -6 \implies x = \frac{3}{2}$

4) $3^{4x} = 9^{x + 1}$

Express 9 as a power of 3: $9 = 3^2$ Thus, the equation becomes: $3^{4x} = (3^2)^{x + 1} \implies 3^{4x} = 3^{2(x + 1)}$ Equating exponents: $4x = 2(x + 1) \implies 4x = 2x + 2 \implies 2x = 2 \implies x = 1$

5) $4^{x + 1} = 64$

Express 4 and 64 as powers of 2: $4 = 2^2 \quad \text{and} \quad 64 = 2^6$ The equation becomes: $(2^2)^{x + 1} = 2^6 \implies 2^{2(x + 1)} = 2^6$ Equating exponents: $2(x + 1) = 6 \implies 2x + 2 = 6 \implies 2x = 4 \implies x = 2$

Final Answers

1. $x = 4$
2. $x = 2$
3. $x = \frac{3}{2}$
4. $x = 1$
5. $x = 2$

Would you like further details or have any questions?

5 Related Questions

1. How do you convert an exponential equation into a base power equation?
2. What are the properties of exponents that make solving these equations easier?
3. How do you handle negative exponents in exponential equations?
4. How do fractional exponents affect the solution of exponential equations?
5. Why is equating the exponents a valid step when the bases are the same?

Tip

When solving exponential equations, always try to express all terms with the same base to simplify the equation!