Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals.\
$19 - \sqrt{-25}$

Explanation:

Step1: Recall the definition of \(i\)

We know that \(i = \sqrt{-1}\), so we can rewrite \(\sqrt{-25}\) as \(\sqrt{25\times(-1)}\).

Step2: Simplify the radical

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 \(\sqrt{25\times(-1)}=\sqrt{25}\times\sqrt{-1}\). Since \(\sqrt{25} = 5\) and \(\sqrt{-1}=i\), then \(\sqrt{-25}=5i\).

Step3: Substitute back into the original expression

The original expression is \(19-\sqrt{-25}\), substituting \(\sqrt{-25}=5i\) into it, we get \(19 - 5i\).

Answer:

\(19 - 5i\)