Simplify the exponential expression \( 48^{\frac{1}{2}} \) \( 2\left(24^{\frac{1}{2}}\right) \) \( 3\left(16^{\frac{1}{2}}\right) \) \( 4\left(3^{\frac{1}{2}}\right) \) \( 16\left(3^{\frac{1}{2}}\right) \)

To simplify \( 48^{\frac{1}{2}} \), we start by rewriting 48 in terms of its prime factors. \( 48 = 16 \times 3 = 2^4 \times 3 \). Therefore, we have: \[ 48^{\frac{1}{2}} = (16 \times 3)^{\frac{1}{2}} = 16^{\frac{1}{2}} \times 3^{\frac{1}{2}} = 4 \times \sqrt{3} \] Now let’s compare this with the provided options: - \( 2(24^{\frac{1}{2}}) \) simplifies to \( 2(2\sqrt{6}) = 4\sqrt{6} \), which isn’t equal to \( 4\sqrt{3} \). - \( 3(16^{\frac{1}{2}}) \) simplifies to \( 3(4) = 12 \), which isn’t equal to \( 4\sqrt{3} \). - \( 4(3^{\frac{1}{2}}) \) simplifies to \( 4\sqrt{3} \), matching our result. - \( 16(3^{\frac{1}{2}}) \) simplifies to \( 16\sqrt{3} \), which again doesn’t match. Thus, the simplified expression is: \( 4\sqrt{3} \).