Question

write in terms of i. simplify your answer as much as possible. \\(sqrt{-80}\\)

Explanation:

Step1: Recall the definition of imaginary unit

We know that \(i = \sqrt{-1}\), so we can rewrite \(\sqrt{-80}\) as \(\sqrt{80\times(-1)}\).

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 for complex numbers), we have \(\sqrt{80\times(-1)}=\sqrt{80}\times\sqrt{-1}\).

Step3: Simplify \(\sqrt{80}\)

We factorize \(80\) as \(16\times5\), so \(\sqrt{80}=\sqrt{16\times5}\). Again using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (with \(a = 16\geq0\) and \(b = 5\geq0\)), we get \(\sqrt{16\times5}=\sqrt{16}\times\sqrt{5}=4\sqrt{5}\).

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

Since \(\sqrt{-1} = i\), then \(\sqrt{80}\times\sqrt{-1}=4\sqrt{5}\times i = 4i\sqrt{5}\).

Answer:

\(4i\sqrt{5}\)