Write \( 4^{3} \sqrt{16} \) using rational exponents, A. \( 4^{\frac{8}{3}} \) B. 16 C. \( 4^{\frac{1}{76}} \) D. \( 4^{16} \)

Let's analyze the expression \( 4^{3} \sqrt{16} \) and determine which of the given options correctly represents it using rational exponents. ### Step 1: Simplify the Expression 1. **Evaluate \( 4^3 \):** \[ 4^3 = 4 \times 4 \times 4 = 64 \] 2. **Simplify \( \sqrt{16} \):** \[ \sqrt{16} = 16^{1/2} = 4 \] (since \( 16 = 4^2 \), so \( 16^{1/2} = (4^2)^{1/2} = 4^{2 \times \frac{1}{2}} = 4^1 = 4 \)) 3. **Multiply the Results:** \[ 4^3 \sqrt{16} = 64 \times 4 = 256 \] Alternatively, using exponents: \[ 4^3 \times 4^1 = 4^{3+1} = 4^4 = 256 \] ### Step 2: Compare with Given Options Let's evaluate each option to see if any equal 256. - **Option A: \( 4^{\frac{8}{3}} \)** \[ 4^{\frac{8}{3}} = (2^2)^{\frac{8}{3}} = 2^{\frac{16}{3}} \approx 2^{5.333} \approx 40.317 \] This does **not** equal 256. - **Option B: 16** \[ 16 = 4^2 \] This does **not** equal 256. - **Option C: \( 4^{\frac{1}{76}} \)** \[ 4^{\frac{1}{76}} \approx 1.017 \] This does **not** equal 256. - **Option D: \( 4^{16} \)** \[ 4^{16} = (2^2)^{16} = 2^{32} = 4,294,967,296 \] This does **not** equal 256. ### Conclusion None of the provided options correctly represent \( 4^{3} \sqrt{16} \) using rational exponents, as the correct representation is \( 4^4 \), which equals 256. Since \( 4^4 \) isn't listed among the options, **none of the given choices are correct**. **Correct Representation:** \[ 4^3 \sqrt{16} = 4^4 = 256 \] However, if you must choose from the provided options, none accurately represent the expression. **Answer:** None of the provided options are correct.