Which expression is equivalent to $$ (\sqrt[3]{6})^{4} $$ ? $$ 6^{\frac{3}{4}} 6^{7} $$ $$ 6^{\frac{4}{3}} 6^{12} $$

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To determine which expression is equivalent to $$ (\sqrt[3]{6})^{4} $$, we need to simplify the given expression and compare it to the options provided.

First, let's simplify $$ (\sqrt[3]{6})^{4} $$:

$$ (\sqrt[3]{6})^{4} = (6^{\frac{1}{3}})^{4} $$

Now, when you raise a power to a power, you multiply the exponents:

$$ (6^{\frac{1}{3}})^{4} = 6^{\frac{1}{3} 	imes 4} $$

$$ 6^{\frac{1}{3} 	imes 4} = 6^{\frac{4}{3}} $$

So, the expression $$ (\sqrt[3]{6})^{4} $$ simplifies to $$ 6^{\frac{4}{3}} $$.

Now let's compare this to the options provided:

1. $$ 6^{\frac{3}{4}} 6^{7} $$
2. $$ 6^{\frac{4}{3}} 6^{12} $$

The correct answer is the one that matches $$ 6^{\frac{4}{3}} $$. None of the options provided match this exactly, but if we consider the properties of exponents, we can combine the terms in the second option:

$$ 6^{\frac{4}{3}} 6^{12} = 6^{\frac{4}{3} + 12} $$

To add the exponents, we need a common denominator:

$$ \frac{4}{3} + 12 = \frac{4}{3} + \frac{36}{3} = \frac{40}{3} $$

So, the second option simplifies to:

$$ 6^{\frac{40}{3}} $$

This is not equivalent to $$ 6^{\frac{4}{3}} $$, so none of the provided options are correct. The correct expression is simply $$ 6^{\frac{4}{3}} $$.
The correct expression is $$ 6^{\frac{4}{3}} $$.

Which expression is equivalent to \( (\sqrt[3]{6})^{4} \) ? \( 6^{\frac{3}{4}} 6^{7} \) \( 6^{\frac{4}{3}} 6^{12} \)

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