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

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 \( f(x) = \sqrt[2]{162^{x}} \).