Question

write in terms of i. simplify your answer as much as possible. \sqrt{-24}

Explanation:

Step1: Recall the definition of \(i\)

We know that \(i = \sqrt{-1}\), so we can rewrite \(\sqrt{-24}\) as \(\sqrt{24\times(-1)}\).

Step2: Use the property of square roots

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\), and here we extend it to complex numbers), we have \(\sqrt{24\times(-1)}=\sqrt{24}\times\sqrt{-1}\).

Step3: Simplify \(\sqrt{24}\)

We can factor \(24\) as \(4\times6\), so \(\sqrt{24}=\sqrt{4\times6}=\sqrt{4}\times\sqrt{6}=2\sqrt{6}\).

Step4: Substitute back \(\sqrt{-1}=i\)

Since \(\sqrt{-1} = i\), we have \(\sqrt{24}\times\sqrt{-1}=2\sqrt{6}\times i = 2i\sqrt{6}\).

Answer:

\(2i\sqrt{6}\)