Which of the following is equivalent to \( 5 \sqrt{12}=2 \sqrt{277} \) a.) \( 4 \sqrt{3} \) b.) \( 3 \sqrt{15} \) c.) \( 5 \sqrt{15} \) d.) \( 2 \sqrt{3} \)

To determine which of the options is equivalent to \( 5 \sqrt{12} = 2 \sqrt{277} \), we will first simplify \( 5 \sqrt{12} \). 1. **Simplify \( \sqrt{12} \)**: \[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \] 2. **Substituting back into the equation**: \[ 5 \sqrt{12} = 5 \cdot 2\sqrt{3} = 10\sqrt{3} \] Now we have: \[ 10\sqrt{3} = 2\sqrt{277} \] Next, we will check each option to see if any of them equal \( 10\sqrt{3} \). - **Option a: \( 4\sqrt{3} \)**: \[ 4\sqrt{3} \neq 10\sqrt{3} \] - **Option b: \( 3\sqrt{15} \)**: \[ 3\sqrt{15} \text{ (not equal to } 10\sqrt{3}\text{)} \] - **Option c: \( 5\sqrt{15} \)**: \[ 5\sqrt{15} \text{ (not equal to } 10\sqrt{3}\text{)} \] - **Option d: \( 2\sqrt{3} \)**: \[ 2\sqrt{3} \neq 10\sqrt{3} \] None of the options seem to match \( 10\sqrt{3} \). Let's check if \( 2\sqrt{277} \) can be simplified or if any of the options can be expressed in terms of \( \sqrt{3} \). 3. **Check \( 2\sqrt{277} \)**: - Since \( 277 \) is a prime number, \( \sqrt{277} \) cannot be simplified further. Since none of the options are equivalent to \( 10\sqrt{3} \) or \( 2\sqrt{277} \), it appears that there may be a misunderstanding in the problem statement or the options provided. Thus, none of the options a, b, c, or d are equivalent to \( 5\sqrt{12} \) or \( 2\sqrt{277} \).