Which expression is equivalent to $$ \sqrt{7^{5}} $$ ? $$ 7^{\frac{2}{5}} 7^{\frac{5}{2}} $$ $$ 7^{\frac{1}{5}} 7^{3} $$

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$$ 7^{\frac{5}{2}} $$
To determine which expression is equivalent to $$ \sqrt{7^{5}} $$, we need to simplify the square root and compare it to the given options.

First, let's simplify $$ \sqrt{7^{5}} $$:

$$ \sqrt{7^{5}} = (7^{5})^{\frac{1}{2}} $$

Using the power rule for exponents, which states that $$ (a^{m})^{n} = a^{m \cdot n} $$, we can simplify further:

$$ (7^{5})^{\frac{1}{2}} = 7^{5 \cdot \frac{1}{2}} $$

$$ 7^{5 \cdot \frac{1}{2}} = 7^{\frac{5}{2}} $$

Now, let's compare this to the given options:

1. $$ 7^{\frac{2}{5}} 7^{\frac{5}{2}} $$
2. $$ 7^{\frac{1}{5}} 7^{3} $$
3. $$ 7^{\frac{5}{2}} $$

The expression $$ 7^{\frac{5}{2}} $$ matches the simplified form of $$ \sqrt{7^{5}} $$, so the correct answer is:

$$ 7^{\frac{5}{2}} $$
$$ 7^{\frac{5}{2}} $$

Which expression is equivalent to ?

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