Which exponential functions have been simplified correctly? Check all that apply. \( f(x)=5 \sqrt[3]{16}^{x}=5(2 \sqrt[3]{2})^{x} \) \( f(x)=2.3(8)^{\frac{1}{2} x}=2.3(4)^{x} \) \( f(x)=81^{\frac{x}{4}}=3^{x} \) \( f(x)=\frac{3}{4} \sqrt{27^{x}}=\frac{3}{4}(3 \sqrt{3})^{x} \) \( f(x)=(24)^{\frac{1}{3} x}=2(\sqrt[3]{3})^{x} \)

Let's analyze each exponential function to determine if it has been simplified correctly. 1. **\( f(x)=5 \sqrt[3]{16}^{x}=5(2 \sqrt[3]{2})^{x} \)** - **Original Expression:** \( 5 \cdot 16^{\frac{1}{3}x} \) - **Simplification:** \( 5 \cdot (2 \cdot 2^{\frac{1}{3}})^x = 5 \cdot (2^{\frac{4}{3}})^x = 5 \cdot 2^{\frac{4}{3}x} \) Since \( 16^{\frac{1}{3}} = 2 \sqrt[3]{2} \), the simplification is **correct**. 2. **\( f(x)=2.3(8)^{\frac{1}{2} x}=2.3(4)^{x} \)** - **Original Expression:** \( 2.3 \cdot 8^{\frac{1}{2}x} = 2.3 \cdot (2 \sqrt{2})^x \) - **Simplification:** \( 2.3 \cdot 4^x \) However, \( 8^{\frac{1}{2}} = 2 \sqrt{2} \), not \( 4 \). Therefore, the simplification is **incorrect**. 3. **\( f(x)=81^{\frac{x}{4}}=3^{x} \)** - **Original Expression:** \( 81^{\frac{x}{4}} = (3^4)^{\frac{x}{4}} = 3^x \) The simplification is **correct**. 4. **\( f(x)=\frac{3}{4} \sqrt{27^{x}}=\frac{3}{4}(3 \sqrt{3})^{x} \)** - **Original Expression:** \( \frac{3}{4} \cdot 27^{\frac{x}{2}} = \frac{3}{4} \cdot (3 \sqrt{3})^x \) Since \( 27^{\frac{1}{2}} = 3 \sqrt{3} \), the simplification is **correct**. 5. **\( f(x)=(24)^{\frac{1}{3} x}=2(\sqrt[3]{3})^{x} \)** - **Original Expression:** \( 24^{\frac{x}{3}} = (2^3 \cdot 3)^{\frac{x}{3}} = 2^x \cdot 3^{\frac{x}{3}} \) - **Simplification:** \( 2 \cdot 3^{\frac{x}{3}} \) These are not equivalent unless \( x = 1 \). Therefore, the simplification is **incorrect**. **In summary, the correctly simplified exponential functions are the first, third, and fourth options.** **Answer:** All except the second and fifth functions are simplified correctly.