Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals.
-9 + \sqrt{-76}

Explanation:

Step1: Recall the definition of imaginary number

We know that \(i = \sqrt{-1}\), so we can rewrite \(\sqrt{-76}\) as \(\sqrt{76\times(-1)}\).
Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\), here we extend it for complex numbers), we have \(\sqrt{76\times(-1)}=\sqrt{76}\times\sqrt{-1}\).

Step2: Simplify \(\sqrt{76}\)

Factor 76: \(76 = 4\times19\), so \(\sqrt{76}=\sqrt{4\times19}\).
Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a = 4,b = 19\), both non - negative), we get \(\sqrt{4\times19}=\sqrt{4}\times\sqrt{19}=2\sqrt{19}\).

Step3: Substitute back

Since \(\sqrt{-76}=\sqrt{76}\times\sqrt{-1}\), and we know that \(\sqrt{-1}=i\) and \(\sqrt{76} = 2\sqrt{19}\), then \(\sqrt{-76}=2\sqrt{19}i\).
The original expression is \(-9+\sqrt{-76}\), substituting \(\sqrt{-76}=2\sqrt{19}i\) into it, we get \(-9 + 2\sqrt{19}i\).

Answer:

\(-9 + 2\sqrt{19}i\)