Which expression is equivalent to $$ 7 f^{\frac{3}{5}} $$ ? $$ 35 \sqrt{f^{3}} \sqrt[5]{7 f^{3}} $$ $$ 7 \sqrt[5]{f^{3}} \sqrt[5]{(7 f)^{3}} $$

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To determine which expression is equivalent to $$ 7 f^{\frac{3}{5}} $$, we will simplify both given expressions and see if they match $$ 7 f^{\frac{3}{5}} $$.

1. **First Expression: $$ 35 \sqrt{f^{3}} \sqrt[5]{7 f^{3}} $$**

We can rewrite the square root and the fifth root:
   $$
   \sqrt{f^{3}} = f^{\frac{3}{2}}
   $$
   $$
   \sqrt[5]{7 f^{3}} = \sqrt[5]{7} \cdot \sqrt[5]{f^{3}} = \sqrt[5]{7} \cdot f^{\frac{3}{5}}
   $$

Now substituting these into the expression:
   $$
   35 \sqrt{f^{3}} \sqrt[5]{7 f^{3}} = 35 f^{\frac{3}{2}} \cdot \sqrt[5]{7} \cdot f^{\frac{3}{5}}
   $$

We can combine the powers of $$ f $$:
   $$
   f^{\frac{3}{2}} \cdot f^{\frac{3}{5}} = f^{\frac{3}{2} + \frac{3}{5}}
   $$

To add the exponents, we need a common denominator. The least common multiple of 2 and 5 is 10:
   $$
   \frac{3}{2} = \frac{15}{10}, \quad \frac{3}{5} = \frac{6}{10}
   $$
   Thus,
   $$
   \frac{3}{2} + \frac{3}{5} = \frac{15}{10} + \frac{6}{10} = \frac{21}{10}
   $$

Therefore,
   $$
   35 f^{\frac{21}{10}} \cdot \sqrt[5]{7}
   $$

This does not simplify to $$ 7 f^{\frac{3}{5}} $$.

2. **Second Expression: $$ 7 \sqrt[5]{f^{3}} \sqrt[5]{(7 f)^{3}} $$**

We can simplify this expression:
   $$
   \sqrt[5]{f^{3}} = f^{\frac{3}{5}}
   $$
   $$
   \sqrt[5]{(7 f)^{3}} = \sqrt[5]{7^{3}} \cdot \sqrt[5]{f^{3}} = 7^{\frac{3}{5}} \cdot f^{\frac{3}{5}}
   $$

Now substituting these into the expression:
   $$
   7 \cdot f^{\frac{3}{5}} \cdot (7^{\frac{3}{5}} \cdot f^{\frac{3}{5}}) = 7 \cdot 7^{\frac{3}{5}} \cdot f^{\frac{3}{5}} \cdot f^{\frac{3}{5}}
   $$

Combining the powers of $$ f $$:
   $$
   f^{\frac{3}{5}} \cdot f^{\frac{3}{5}} = f^{\frac{6}{5}}
   $$

Thus, we have:
   $$
   7 \cdot 7^{\frac{3}{5}} \cdot f^{\frac{6}{5}} = 7^{1 + \frac{3}{5}} \cdot f^{\frac{6}{5}} = 7^{\frac{8}{5}} \cdot f^{\frac{6}{5}}
   $$

This also does not simplify to $$ 7 f^{\frac{3}{5}} $$.

After evaluating both expressions, neither of them is equivalent to $$ 7 f^{\frac{3}{5}} $$.

Thus, the answer is that neither expression is equivalent to $$ 7 f^{\frac{3}{5}} $$.
Neither expression is equivalent to $$ 7 f^{\frac{3}{5}} $$.

Which expression is equivalent to ?

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