Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals. \\(\sqrt{-4}\\)

Explanation:

Step1: Recall the definition of \(i\)

The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\). So, we can rewrite \(\sqrt{-4}\) by factoring out \(\sqrt{-1}\) from the radical.
\(\sqrt{-4}=\sqrt{4\times(-1)}\)

Step2: Use the property of square roots

We know that \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) for \(a\geq0\) and \(b\) as a real number (in this case, \(a = 4\) and \(b=- 1\)). So,
\(\sqrt{4\times(-1)}=\sqrt{4}\times\sqrt{-1}\)
Since \(\sqrt{4} = 2\) and \(\sqrt{-1}=i\), we substitute these values in:
\(\sqrt{4}\times\sqrt{-1}=2\times i = 2i\)

Answer:

\(2i\)