Question

question
write \\(\sqrt{-54}\\) in simplest radical form.
answer attempt 1 out of 2

Explanation:

Step1: Separate the negative sign and the number

We know that \(\sqrt{-a}=\sqrt{-1}\cdot\sqrt{a}\) (for \(a\geq0\)). So, \(\sqrt{-54}=\sqrt{-1}\cdot\sqrt{54}\).

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

Factor \(54\) into perfect square and non - perfect square factors. \(54 = 9\times6\), and \(\sqrt{9\times6}=\sqrt{9}\cdot\sqrt{6}\) (by the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\), where \(a = 9\), \(b = 6\) and \(a\geq0\), \(b\geq0\)). Since \(\sqrt{9}=3\), then \(\sqrt{54}=3\sqrt{6}\).

Step3: Recall the definition of \(i\)

We know that \(i=\sqrt{-1}\). So, \(\sqrt{-1}\cdot\sqrt{54}=i\cdot3\sqrt{6}\).

Step4: Combine the terms

Rearranging the terms, we get \(3i\sqrt{6}\).

Answer:

\(3i\sqrt{6}\)