5 Select ALL the correct answers. Which of the following properties can be used to show that the expression $$ 4 \frac{\frac{5}{5}}{} $$ is equivalent to $$ \sqrt[3]{4^{5}} $$ ? $$ 4^{\frac{8}{2}} \cdot 4^{\frac{7}{3}}=4^{\left(\frac{1}{3}+\frac{7}{3}\right)}=4^{5} $$ $$ \frac{4^{\frac{17}{3}}}{4^{\frac{1}{3}}}=4^{\left(\frac{17}{3}-\frac{2}{3}\right)}=4^{\mathrm{E}} $$ $$ \sqrt[2]{4^{5}}=\left(4^{5}\right)^{\frac{1}{3}}=4^{\frac{6}{3}} $$ $$ \left(4^{15}\right)^{\frac{1}{2}}=4^{\left(15-\frac{1}{1}\right)}=4^{5} $$ $$ \left(4^{\frac{3}{3}}\right)^{3}=4^{\left(\frac{3}{3} \cdot 3\right)}=4^{5} $$

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To determine which properties can be used to show that the expression $$ 4 \frac{\frac{5}{5}}{} $$ is equivalent to $$ \sqrt[3]{4^{5}} $$, we need to analyze each option step by step.

1. **Understanding the expressions:**
   - The expression $$ 4 \frac{\frac{5}{5}}{} $$ simplifies to $$ 4 $$ since $$ \frac{5}{5} = 1 $$.
   - The expression $$ \sqrt[3]{4^{5}} $$ can be rewritten using the property of exponents: $$ \sqrt[3]{4^{5}} = 4^{\frac{5}{3}} $$.

2. **Comparing the two expressions:**
   - We need to check if $$ 4 $$ (or $$ 4^{1} $$) is equivalent to $$ 4^{\frac{5}{3}} $$. Clearly, $$ 4^{1} \neq 4^{\frac{5}{3}} $$, so we need to find properties that can help us show equivalence.

Now, let's analyze each option:

1. **Option 1:** 
   $$
   4^{\frac{8}{2}} \cdot 4^{\frac{7}{3}}=4^{\left(\frac{8}{2}+\frac{7}{3}\right)}=4^{5}
   $$
   - This is correct because $$ \frac{8}{2} = 4 $$ and $$ \frac{7}{3} $$ can be added to get $$ 4 + \frac{7}{3} $$, which does not equal $$ 5 $$. Thus, this option is **not correct**.

2. **Option 2:**
   $$
   \frac{4^{\frac{17}{3}}}{4^{\frac{1}{3}}}=4^{\left(\frac{17}{3}-\frac{1}{3}\right)}=4^{\frac{16}{3}}
   $$
   - This simplifies to $$ 4^{\frac{16}{3}} $$, which is not equal to $$ 4^{5} $$. Thus, this option is **not correct**.

3. **Option 3:**
   $$
   \sqrt[2]{4^{5}}=\left(4^{5}\right)^{\frac{1}{3}}=4^{\frac{5}{2}}
   $$
   - This simplifies to $$ 4^{\frac{5}{2}} $$, which is not equal to $$ 4^{5} $$. Thus, this option is **not correct**.

4. **Option 4:**
   $$
   \left(4^{15}\right)^{\frac{1}{2}}=4^{\left(15 \cdot \frac{1}{2}\right)}=4^{\frac{15}{2}}
   $$
   - This simplifies to $$ 4^{\frac{15}{2}} $$, which is not equal to $$ 4^{5} $$. Thus, this option is **not correct**.

5. **Option 5:**
   $$
   \left(4^{\frac{3}{3}}\right)^{3}=4^{\left(\frac{3}{3} \cdot 3\right)}=4^{3}
   $$
   - This simplifies to $$ 4^{3} $$, which is not equal to $$ 4^{5} $$. Thus, this option is **not correct**.

After analyzing all options, none of the provided properties can be used to show that $$ 4 $$ is equivalent to $$ \sqrt[3]{4^{5}} $$. Therefore, the answer is that **none of the options are correct**.
None of the provided properties correctly show that $$ 4 $$ is equivalent to $$ \sqrt[3]{4^{5}} $$.

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