Simpllly the radical. sqrt (-48) square

To simplify the radical $\sqrt{-48}$, we first need to recognize that the square root of a negative number involves the imaginary unit $i$, where $i = \sqrt{-1}$. <br /><br />Given:<br />\[<br />\sqrt{-48}<br />\]<br /><br />We can rewrite $-48$ as:<br />\[<br />-48 = -1 \times 48<br />\]<br /><br />Thus:<br />\[<br />\sqrt{-48} = \sqrt{-1 \times 48}<br />\]<br /><br />Since $\sqrt{-1} = i$, we have:<br />\[<br />\sqrt{-48} = \sqrt{48} \times \sqrt{-1} = \sqrt{48} \times i<br />\]<br /><br />Next, we simplify $\sqrt{48}$. The number 48 can be factored into its prime factors:<br />\[<br />48 = 16 \times 3 = 2^4 \times 3<br />\]<br /><br />Therefore:<br />\[<br />\sqrt{48} = \sqrt{2^4 \times 3} = \sqrt{2^4} \times \sqrt{3} = 4\sqrt{3}<br />\]<br /><br />Combining this with the imaginary unit $i$, we get:<br />\[<br />\sqrt{-48} = 4\sqrt{3} \times i = 4i\sqrt{3}<br />\]<br /><br />So, the simplified form of the radical $\sqrt{-48}$ is:<br />\[<br />\boxed{4i\sqrt{3}}<br />\]