Which is equivalent to \( 80^{\frac{1}{4} x} \) ? \( \left(\frac{80}{4}\right)^{x} \) \( \sqrt[4]{80} x \) \( \sqrt[x]{80} \) \( \left(\frac{80}{x}\right)^{4} \)

To determine which expression is equivalent to \( 80^{\frac{1}{4} x} \), let's analyze each option step by step. ### Given Expression: \[ 80^{\frac{1}{4} x} \] This can be rewritten using the properties of exponents: \[ 80^{\frac{x}{4}} = \left(80^{\frac{1}{4}}\right)^x = \left(\sqrt[4]{80}\right)^x \] ### Evaluating the Options: 1. **\( \left(\frac{80}{4}\right)^{x} \)** \[ \left(\frac{80}{4}\right)^x = 20^x \] - **Comparison:** \( 80^{\frac{x}{4}} \) vs. \( 20^x \) - These are **not** equivalent because \( 80^{\frac{x}{4}} \) is not equal to \( 20^x \) for general \( x \). 2. **\( \sqrt[4]{80} \times x \)** \[ \sqrt[4]{80} \times x = 80^{\frac{1}{4}} \times x \] - **Comparison:** \( 80^{\frac{x}{4}} \) vs. \( 80^{\frac{1}{4}} \times x \) - These are **not** equivalent because one involves an exponential function of \( x \), while the other is a linear function of \( x \). 3. **\( \sqrt[x]{80} \)** \[ \sqrt[x]{80} = 80^{\frac{1}{x}} \] - **Comparison:** \( 80^{\frac{x}{4}} \) vs. \( 80^{\frac{1}{x}} \) - These are **not** equivalent unless \( x = \frac{1}{4} \), which isn’t generally true. 4. **\( \left(\frac{80}{x}\right)^{4} \)** \[ \left(\frac{80}{x}\right)^4 = \frac{80^4}{x^4} \] - **Comparison:** \( 80^{\frac{x}{4}} \) vs. \( \frac{80^4}{x^4} \) - These are **not** equivalent for general \( x \). ### Correct Equivalent Expression: None of the provided options exactly match the equivalent expression \( 80^{\frac{x}{4}} = \left(\sqrt[4]{80}\right)^x \). However, if we were to adjust **Option 2** to include the exponent properly, it would become equivalent: - **Adjusted Option 2:** \[ \left(\sqrt[4]{80}\right)^x \] This matches our derived equivalent expression. It seems there might have been a slight misrepresentation in **Option 2**, where the exponent was intended but omitted. ### Conclusion: **None of the given options exactly match \( 80^{\frac{1}{4} x} \).** The closest equivalent expression would be \( \left(\sqrt[4]{80}\right)^x \), which isn't precisely listed among the options provided.