Which expression is equivalent to $$ 4^{\frac{5}{3}} $$ ? $$ \sqrt[4]{3^{5}} \sqrt[5]{3^{4}} $$ $$ \sqrt[5]{4^{3}} \sqrt[3]{4^{5}} $$

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To find an expression equivalent to $$ 4^{\frac{5}{3}} $$, we can simplify the two given options and see if either matches $$ 4^{\frac{5}{3}} $$.

1. **First expression: $$ \sqrt[4]{3^{5}} \sqrt[5]{3^{4}} $$**

We can rewrite this expression using exponent notation:
   $$
   \sqrt[4]{3^{5}} = 3^{\frac{5}{4}} \quad \text{and} \quad \sqrt[5]{3^{4}} = 3^{\frac{4}{5}}
   $$
   Therefore, we have:
   $$
   \sqrt[4]{3^{5}} \sqrt[5]{3^{4}} = 3^{\frac{5}{4}} \cdot 3^{\frac{4}{5}} = 3^{\frac{5}{4} + \frac{4}{5}}
   $$
   To combine the exponents, we need a common denominator. The least common multiple of 4 and 5 is 20:
   $$
   \frac{5}{4} = \frac{25}{20} \quad \text{and} \quad \frac{4}{5} = \frac{16}{20}
   $$
   Thus,
   $$
   \frac{5}{4} + \frac{4}{5} = \frac{25}{20} + \frac{16}{20} = \frac{41}{20}
   $$
   So, the first expression simplifies to:
   $$
   3^{\frac{41}{20}}
   $$

2. **Second expression: $$ \sqrt[5]{4^{3}} \sqrt[3]{4^{5}} $$**

Again, we can rewrite this using exponent notation:
   $$
   \sqrt[5]{4^{3}} = 4^{\frac{3}{5}} \quad \text{and} \quad \sqrt[3]{4^{5}} = 4^{\frac{5}{3}}
   $$
   Therefore, we have:
   $$
   \sqrt[5]{4^{3}} \sqrt[3]{4^{5}} = 4^{\frac{3}{5}} \cdot 4^{\frac{5}{3}} = 4^{\frac{3}{5} + \frac{5}{3}}
   $$
   To combine the exponents, we again need a common denominator. The least common multiple of 5 and 3 is 15:
   $$
   \frac{3}{5} = \frac{9}{15} \quad \text{and} \quad \frac{5}{3} = \frac{25}{15}
   $$
   Thus,
   $$
   \frac{3}{5} + \frac{5}{3} = \frac{9}{15} + \frac{25}{15} = \frac{34}{15}
   $$
   So, the second expression simplifies to:
   $$
   4^{\frac{34}{15}}
   $$

Now we compare the simplified forms:

- The first expression simplifies to $$ 3^{\frac{41}{20}} $$.
- The second expression simplifies to $$ 4^{\frac{34}{15}} $$.

Neither of these expressions is equal to $$ 4^{\frac{5}{3}} $$.

Thus, the answer is that neither expression is equivalent to $$ 4^{\frac{5}{3}} $$.
Neither expression is equivalent to $$ 4^{\frac{5}{3}} $$.

Which expression is equivalent to ?

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