Question

question 1
1 pts
simplify: \\(\sqrt{-48}\\)
\\(\bigcirc\\) cant be done
\\(\bigcirc\\) \\(4\sqrt{3}i\\)
\\(\bigcirc\\) -48i
\\(\bigcirc\\) \\(-\sqrt{48}\\)

Explanation:

Step1: Recall imaginary unit definition

Recall that \( \sqrt{-a}=\sqrt{a}\cdot i \) for \( a>0 \), where \( i \) is the imaginary unit with \( i^2 = - 1 \). So, \( \sqrt{-48}=\sqrt{48}\cdot i \).

Step2: Simplify \( \sqrt{48} \)

Factor 48 into perfect - square and non - perfect - square factors: \( 48 = 16\times3 \). Then \( \sqrt{48}=\sqrt{16\times3} \).
Using the property \( \sqrt{ab}=\sqrt{a}\cdot\sqrt{b} \) (for \( a\geq0,b\geq0 \)), we have \( \sqrt{16\times3}=\sqrt{16}\cdot\sqrt{3} \).
Since \( \sqrt{16} = 4 \), then \( \sqrt{48}=4\sqrt{3} \).

Step3: Combine with imaginary unit

Substitute \( \sqrt{48}=4\sqrt{3} \) back into \( \sqrt{-48}=\sqrt{48}\cdot i \), we get \( \sqrt{-48}=4\sqrt{3}i \).

Answer:

\( 4\sqrt{3}i \) (corresponding to the option " \( 4\sqrt{3}i \)")