1. **State the problem:** We need to determine which given expressions are equivalent to the expressions they are compared to. 2. **Recall exponent rules:** - $a^{m} \times a^{n} = a^{m+n}$ - $\frac{a^{m}}{a^{n}} = a^{m-n}$ - $(a^{m})^{n} = a^{m \times n}$ - $a^{0} = 1$ 3. **Check each statement:** - Statement 1: $(2^{1/2} \times x) / (2^{3y})$ vs $2^{3(y-x)}$ Rewrite numerator: $2^{1/2} \times x$ is not an exponent expression with base 2 only because of $x$ multiplication. The right side is $2^{3(y-x)} = 2^{3y - 3x}$. The left side is $\frac{2^{1/2} \times x}{2^{3y}} = x \times 2^{1/2 - 3y}$. Since $x$ is multiplied outside the power, it is not equivalent to $2^{3(y-x)}$ which is a pure power of 2. **Statement 1 is false.** - Statement 2: $8^{3x - 1/3}$ vs $5^{12x} / 2$ Rewrite $8 = 2^{3}$, so $8^{3x - 1/3} = (2^{3})^{3x - 1/3} = 2^{9x - 1}$. Right side is $\frac{5^{12x}}{2}$. Bases differ (2 vs 5), so expressions are not equivalent. **Statement 2 is false.** - Statement 3: $6^{y}$ vs $\frac{6x}{6y}$ Right side simplifies to $\frac{6x}{6y} = \frac{x}{y}$. Left side is an exponential expression, right side is a ratio of variables. Not equivalent. **Statement 3 is false.** - Statement 4: $0.5^{2x + 3}$ vs $(0.25)^{x + 3}$ Rewrite $0.5 = \frac{1}{2} = 2^{-1}$ and $0.25 = \frac{1}{4} = 2^{-2}$. Left: $0.5^{2x + 3} = (2^{-1})^{2x + 3} = 2^{-2x - 3}$. Right: $(0.25)^{x + 3} = (2^{-2})^{x + 3} = 2^{-2x - 6}$. Since $2^{-2x - 3} \neq 2^{-2x - 6}$, they are not equivalent. **Statement 4 is false.** - Statement 5: $5^{2x - y}$ vs $25^{x} \times 5^{y}$ Rewrite $25 = 5^{2}$. Right side: $25^{x} \times 5^{y} = (5^{2})^{x} \times 5^{y} = 5^{2x} \times 5^{y} = 5^{2x + y}$. Left side is $5^{2x - y}$. Since $5^{2x - y} \neq 5^{2x + y}$, they are not equivalent. **Statement 5 is false.** **Final answer: None of the statements are true.**