Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals.
$-sqrt{-16}$

Explanation:

Step1: Recall the definition of \(i\)

We know that \(i = \sqrt{-1}\), so we can rewrite \(\sqrt{-16}\) as \(\sqrt{16\times(-1)}\).
Using the property of square roots \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\), here we extend it to complex numbers), we have \(\sqrt{16\times(-1)}=\sqrt{16}\times\sqrt{-1}\).
Since \(\sqrt{16} = 4\) and \(\sqrt{-1}=i\), then \(\sqrt{-16}=4i\).

Step2: Apply the negative sign

The original expression is \(-\sqrt{-16}\), so substituting \(\sqrt{-16}=4i\) into it, we get \(- 4i\).

Answer:

\(-4i\)