Question

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

Explanation:

Step 1: Recall the definition of \(i\)

We know that \(i = \sqrt{-1}\), so we can rewrite \(\sqrt{-20}\) as \(\sqrt{20\times(-1)}\).
Using the property of square roots \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\), and here we extend it for complex numbers), we get \(\sqrt{20}\times\sqrt{-1}\).

Step 2: Simplify \(\sqrt{20}\)

We factor \(20\) as \(4\times5\), so \(\sqrt{20}=\sqrt{4\times5}\).
Again using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (with \(a = 4\) and \(b=5\), both non - negative), we have \(\sqrt{4}\times\sqrt{5}=2\sqrt{5}\).

Step 3: Substitute back \(\sqrt{-1}=i\)

Since \(\sqrt{-20}=\sqrt{20}\times\sqrt{-1}\), and we found that \(\sqrt{20} = 2\sqrt{5}\) and \(\sqrt{-1}=i\), then \(\sqrt{-20}=2\sqrt{5}i\) or \(2i\sqrt{5}\).

Answer:

\(2i\sqrt{5}\) (or \(2\sqrt{5}i\))