Which functions are equivalent to $$ f(x)=\sqrt[2]{162^{x}} $$ ? Check all that apply. $$ f(x)=162^{\frac{x}{4}} $$ $$ f(x)=(3 \sqrt[3]{2})^{x} $$ $$ f(x)=9 \sqrt[4]{2^{x}} $$ $$ f(x)=162^{\frac{4}{x}} $$ $$ f(x)=\left[3\left(2^{\frac{1}{4}}\right)\right]^{x} $$

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To determine which functions are equivalent to $$ f(x) = \sqrt[2]{162^{x}} $$, we can simplify $$ f(x) $$ and then compare it with the given options.

1. **Simplifying $$ f(x) $$**:
   $$
   f(x) = \sqrt[2]{162^{x}} = (162^{x})^{\frac{1}{2}} = 162^{\frac{x}{2}}
   $$

Now we have $$ f(x) = 162^{\frac{x}{2}} $$.

2. **Now, let's analyze each option**:

- **Option 1**: $$ f(x) = 162^{\frac{x}{4}} $$
     - This is not equivalent because $$ \frac{x}{4} \neq \frac{x}{2} $$.

- **Option 2**: $$ f(x) = (3 \sqrt[3]{2})^{x} $$
     - We need to express $$ 162 $$ in terms of its prime factors:
       $$
       162 = 2 \cdot 81 = 2 \cdot 3^4
       $$
       Thus,
       $$
       162^{\frac{x}{2}} = (2 \cdot 3^4)^{\frac{x}{2}} = 2^{\frac{x}{2}} \cdot 3^{2x}
       $$
       Now, let's check $$ (3 \sqrt[3]{2})^{x} $$:
       $$
       (3 \sqrt[3]{2})^{x} = 3^{x} \cdot (2^{\frac{1}{3}})^{x} = 3^{x} \cdot 2^{\frac{x}{3}}
       $$
       This is not equivalent to $$ 162^{\frac{x}{2}} $$.

- **Option 3**: $$ f(x) = 9 \sqrt[4]{2^{x}} $$
     - We can express $$ 9 $$ as $$ 3^2 $$:
       $$
       9 \sqrt[4]{2^{x}} = 3^2 \cdot (2^{\frac{x}{4}}) = 3^2 \cdot 2^{\frac{x}{4}}
       $$
       This is not equivalent to $$ 162^{\frac{x}{2}} $$.

- **Option 4**: $$ f(x) = 162^{\frac{4}{x}} $$
     - This is not equivalent because $$ \frac{4}{x} \neq \frac{x}{2} $$.

- **Option 5**: $$ f(x) = \left[3\left(2^{\frac{1}{4}}\right)\right]^{x} $$
     - This can be simplified:
       $$
       \left[3\left(2^{\frac{1}{4}}\right)\right]^{x} = 3^{x} \cdot (2^{\frac{1}{4}})^{x} = 3^{x} \cdot 2^{\frac{x}{4}}
       $$
       This is not equivalent to $$ 162^{\frac{x}{2}} $$.

3. **Conclusion**:
None of the provided functions are equivalent to $$ f(x) = \sqrt[2]{162^{x}} $$.
None of the given functions are equivalent to $$ f(x) = \sqrt[2]{162^{x}} $$.

Which functions are equivalent to \( f(x)=\sqrt[2]{162^{x}} \) ? Check all that apply. \( f(x)=162^{\frac{x}{4}} \) \( f(x)=(3 \sqrt[3]{2})^{x} \) \( f(x)=9 \sqrt[4]{2^{x}} \) \( f(x)=162^{\frac{4}{x}} \) \( f(x)=\left[3\left(2^{\frac{1}{4}}\right)\right]^{x} \)

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