Let's solve these equations one by one. Most of these problems involve manipulating exponential equations.

1) $16^{2x - 3} = 4^{x + 2}$

Rewrite both sides with the same base (since $16 = 4^2$): $(4^2)^{2x - 3} = 4^{x + 2}$ Simplifying: $4^{4x - 6} = 4^{x + 2}$ Equating the exponents: $4x - 6 = x + 2$ Solve for $x$: $4x - x = 2 + 6 \quad \Rightarrow \quad 3x = 8 \quad \Rightarrow \quad x = \frac{8}{3}$

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

Rewrite the left-hand side: $2^{-2x} = 2^{3 - x}$ Equating the exponents: $-2x = 3 - x$ Solve for $x$: $-2x + x = 3 \quad \Rightarrow \quad -x = 3 \quad \Rightarrow \quad x = -3$

3) $4^{2x + 7} \leq 32^{2x - 3}$

Rewrite $32$ as $2^5$ and $4$ as $2^2$: $(2^2)^{2x + 7} \leq (2^5)^{2x - 3}$ Simplify: $2^{4x + 14} \leq 2^{10x - 15}$ Equate the exponents: $4x + 14 \leq 10x - 15$ Solve for $x$: $14 + 15 \leq 10x - 4x \quad \Rightarrow \quad 29 \leq 6x \quad \Rightarrow \quad x \geq \frac{29}{6}$

4) $\left( \frac{2}{5} \right)^{5x - 1} \geq \frac{25}{4}$

Rewrite $\frac{25}{4}$ as $\left( \frac{5}{2} \right)^2$: $\left( \frac{2}{5} \right)^{5x - 1} \geq \left( \frac{5}{2} \right)^2$ Invert the base of the left side to match the right: $\left( \frac{5}{2} \right)^{-(5x - 1)} \geq \left( \frac{5}{2} \right)^2$ Equate the exponents: $-(5x - 1) \geq 2 \quad \Rightarrow \quad -5x + 1 \geq 2 \quad \Rightarrow \quad -5x \geq 1 \quad \Rightarrow \quad x \leq -\frac{1}{5}$

5) $\left( \frac{2}{3} \right)^{5x + 2} = \left( \frac{3}{2} \right)^{2x}$

Invert the right-hand side: $\left( \frac{2}{3} \right)^{5x + 2} = \left( \frac{2}{3} \right)^{-2x}$ Equate the exponents: $5x + 2 = -2x$ Solve for $x$: $5x + 2x = -2 \quad \Rightarrow \quad 7x = -2 \quad \Rightarrow \quad x = -\frac{2}{7}$

I can continue solving the remaining problems (6-10). Would you like me to do that or explain any specific steps? Here are five relative questions to enhance your understanding:

1. How can you simplify exponential equations with the same base?
2. What strategies work for solving inequalities involving exponents?
3. Why is it necessary to express numbers with the same base in these equations?
4. How can negative exponents affect solving equations?
5. Can you apply similar principles to logarithmic equations?

Tip:

When solving exponential equations, it's often easier to rewrite terms with the same base for direct comparison.