Question 1-13 Select all the true statements. The expression $$ \frac{2^{\frac{1}{3}}}{2^{3 y}} $$ is equivaient to $$ 2^{3(y-x)} $$. The expression $$ 8^{3 x-\frac{1}{3}} $$ is equivalent to $$ \frac{512^{x}}{2} $$ The expression $$ 6^{\frac{x}{7}} $$ is equivalent to $$ \frac{6 x}{6 y} $$. The expression $$ 0.5^{2 x+3} $$ is equivalent to $$ (0.25)^{2+3} $$ The expression $$ 5^{2 x+1} $$ is equivalent to $$ 25^{x} \cdot 5^{y} $$.

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Let's evaluate each of the given statements to determine which are true.

1. **$$ \frac{2^{\frac{1}{3}}}{2^{3 y}} $$ is equivalent to $$ 2^{3(y-x)} $$.**

**Analysis:**
   $$
   \frac{2^{\frac{1}{3}}}{2^{3y}} = 2^{\frac{1}{3} - 3y}
   $$
   For this to be equivalent to $$ 2^{3(y-x)} $$, we would need:
   $$
   \frac{1}{3} - 3y = 3(y - x)
   $$
   Simplifying:
   $$
   \frac{1}{3} - 3y = 3y - 3x \implies \frac{1}{3} = 6y - 3x
   $$
   This equality holds only for specific values of $$ x $$ and $$ y $$, not generally. **Thus, this statement is false.**

2. **$$ 8^{3 x-\frac{1}{3}} $$ is equivalent to $$ \frac{512^{x}}{2} $$.**

**Analysis:**
   $$
   8^{3x - \frac{1}{3}} = (2^3)^{3x - \frac{1}{3}} = 2^{9x - 1}
   $$
   $$
   \frac{512^{x}}{2} = \frac{(2^9)^x}{2} = \frac{2^{9x}}{2} = 2^{9x - 1}
   $$
   Both expressions simplify to $$ 2^{9x - 1} $$. **Thus, this statement is true.**

3. **$$ 6^{\frac{x}{7}} $$ is equivalent to $$ \frac{6 x}{6 y} $$.**

**Analysis:**
   The left side is an exponential expression, while the right side simplifies to $$ \frac{x}{y} $$, a rational expression. These are fundamentally different types of expressions and are not equivalent unless under specific conditions, which are not generally true. **Thus, this statement is false.**

4. **$$ 0.5^{2 x+3} $$ is equivalent to $$ (0.25)^{2+3} $$.**

**Analysis:**
   $$
   0.5^{2x + 3}
   $$
   $$
   (0.25)^5 = (0.5^2)^5 = 0.5^{10}
   $$
   For these to be equivalent:
   $$
   2x + 3 = 10 \implies x = \frac{7}{2}
   $$
   This equality holds only for $$ x = \frac{7}{2} $$, not generally. **Thus, this statement is false.**

5. **$$ 5^{2 x+1} $$ is equivalent to $$ 25^{x} \cdot 5^{y} $$.**

**Analysis:**
   $$
   25^{x} \cdot 5^{y} = (5^2)^x \cdot 5^{y} = 5^{2x} \cdot 5^{y} = 5^{2x + y}
   $$
   For this to be equivalent to $$ 5^{2x + 1} $$:
   $$
   2x + y = 2x + 1 \implies y = 1
   $$
   This equality holds only if $$ y = 1 $$, not generally. **Thus, this statement is false.**

**Conclusion:**

Only the **second statement** is **true**.

**Answer:**  
Only the second statement is true.
Only the second statement is true.

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