Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals.
$-sqrt{-54}$

Explanation:

Step1: Rewrite the negative radicand

Recall that \(\sqrt{-a}=\sqrt{a}\cdot\sqrt{-1}\) for \(a\geq0\), and \(\sqrt{-1} = i\). So, we can rewrite \(\sqrt{-54}\) as \(\sqrt{54}\cdot\sqrt{-1}\).
\(-\sqrt{-54}=-\sqrt{54}\cdot\sqrt{-1}\)

Step2: Simplify the radical \(\sqrt{54}\)

Factor \(54\) into its prime factors: \(54 = 9\times6\), and \(\sqrt{9\times6}=\sqrt{9}\times\sqrt{6}\) (by the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) for \(a\geq0,b\geq0\)). Since \(\sqrt{9} = 3\), we have \(\sqrt{54}=3\sqrt{6}\).

Step3: Substitute back and use \(i\)

We know that \(\sqrt{-1}=i\), so substituting \(\sqrt{54} = 3\sqrt{6}\) and \(\sqrt{-1}=i\) into the expression from Step1:
\(-\sqrt{-54}=-(3\sqrt{6})\cdot i=- 3i\sqrt{6}\) (or \(-3\sqrt{6}i\))

Answer:

\(-3\sqrt{6}i\) (or \(-3i\sqrt{6}\))