Question

question
write \\(\sqrt{-54}\\) in simplest radical form.
answer attempt 1 out of 4

Explanation:

Step1: Recall the imaginary unit

Recall that \( \sqrt{-1} = i \). So we can rewrite \( \sqrt{-54} \) as \( \sqrt{54 \times (-1)} \). By the property of square roots \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a\geq0, b\geq0 \), here we extend it for complex numbers), we have \( \sqrt{54\times(-1)}=\sqrt{54}\times\sqrt{-1} \).

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

Factor 54 into its prime factors: \( 54 = 9\times6 \), and \( 9 = 3^2 \). So \( \sqrt{54}=\sqrt{9\times6}=\sqrt{9}\times\sqrt{6}=3\sqrt{6} \).

Step3: Combine with the imaginary unit

Since \( \sqrt{-1}=i \), we substitute back: \( \sqrt{54}\times\sqrt{-1}=3\sqrt{6}\times i = 3i\sqrt{6} \).

Answer:

\( 3i\sqrt{6} \)