Write in terms of i. Simplify. your answer as much as possible. -sqrt (-75) square

To simplify the expression \(-\sqrt{-75}\), we need to express it in terms of the imaginary unit \(i\), where \(i = \sqrt{-1}\).<br /><br />First, rewrite the expression inside the square root:<br />\[<br />-\sqrt{-75} = -\sqrt{75 \cdot (-1)}<br />\]<br /><br />Since \(\sqrt{-1} = i\), we can substitute \(i\) for \(\sqrt{-1}\):<br />\[<br />-\sqrt{75 \cdot (-1)} = -\sqrt{75} \cdot \sqrt{-1} = -\sqrt{75} \cdot i<br />\]<br /><br />Next, simplify \(\sqrt{75}\). We know that:<br />\[<br />75 = 25 \cdot 3<br />\]<br />and<br />\[<br />\sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}<br />\]<br /><br />Therefore, the expression becomes:<br />\[<br />-\sqrt{75} \cdot i = -5\sqrt{3} \cdot i<br />\]<br /><br />So, the simplified form of \(-\sqrt{-75}\) is:<br />\[<br />-\sqrt{-75} = -5\sqrt{3}i<br />\]<br /><br />Thus, the final answer is:<br />\[<br />\boxed{-5\sqrt{3}i}<br />\]