Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals. \\(\sqrt{-20}\\)

Explanation:

Step1: Recall the definition of \(i\)

We know that \(i = \sqrt{-1}\), so we can rewrite \(\sqrt{-20}\) as \(\sqrt{20\times(-1)}\).

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 \(\sqrt{20\times(-1)}=\sqrt{20}\times\sqrt{-1}\).

Step3: Simplify \(\sqrt{20}\)

Simplify \(\sqrt{20}\), we can factor 20 as \(4\times5\), so \(\sqrt{20}=\sqrt{4\times5}=\sqrt{4}\times\sqrt{5} = 2\sqrt{5}\).

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

Since \(\sqrt{-1}=i\), then \(\sqrt{20}\times\sqrt{-1}=2\sqrt{5}\times i = 2i\sqrt{5}\).

Answer:

\(2i\sqrt{5}\)