The image shows three exponential equations. Let me solve them one by one.

1. $7^{x+4} = 49^{2x-1}$

We can rewrite $49$ as $7^2$: $49^{2x-1} = (7^2)^{2x-1} = 7^{2(2x-1)} = 7^{4x - 2}$

Now the equation becomes: $7^{x+4} = 7^{4x-2}$

Since the bases are the same, we equate the exponents: $x + 4 = 4x - 2$

Solving for $x$: $4 + 2 = 4x - x \implies 6 = 3x \implies x = 2$

2. $\left( \frac{1}{2} \right)^{2x} = 2^{3-x}$

We rewrite $\left( \frac{1}{2} \right)^{2x}$ as $2^{-2x}$: $2^{-2x} = 2^{3 - x}$

Since the bases are the same, we equate the exponents: $-2x = 3 - x$

Solving for $x$: $-2x + x = 3 \implies -x = 3 \implies x = -3$

3. $4^{x+2} = 8^{2x}$

We rewrite $4$ and $8$ with base 2: $4 = 2^2 \quad \text{and} \quad 8 = 2^3$

Thus, the equation becomes: $(2^2)^{x+2} = (2^3)^{2x}$

Simplifying both sides: $2^{2(x+2)} = 2^{6x}$

Equate the exponents: $2(x + 2) = 6x$

Simplifying: $2x + 4 = 6x \implies 4 = 4x \implies x = 1$

Final Answers:

1. $x = 2$
2. $x = -3$
3. $x = 1$

Let me know if you want any further explanations or details.

Here are 5 related questions for practice:

1. Solve $5^{x+1} = 25^{2x-3}$.
2. Solve $3^{2x} = 9^{x+1}$.
3. Solve $2^{x+5} = 16^{x-2}$.
4. Solve $(1/3)^{2x} = 3^{x-1}$.
5. Solve $6^{x+2} = 36^{x-3}$.

Tip: Always try to express the numbers in exponential equations with the same base for easier simplification.