Question

question
write \\(\sqrt{-147}\\) 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 \( \sqrt{-147} \) as \( \sqrt{147 \times (-1)}=\sqrt{147}\times\sqrt{-1} \).

Step2: Simplify \( \sqrt{147} \)

Factor 147: \( 147 = 49\times3 \), and since \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a\geq0,b\geq0 \)), we have \( \sqrt{147}=\sqrt{49\times3}=\sqrt{49}\times\sqrt{3} \).
We know that \( \sqrt{49} = 7 \), so \( \sqrt{147}=7\sqrt{3} \).

Step3: Combine with the imaginary unit

From Step 1, \( \sqrt{-147}=\sqrt{147}\times\sqrt{-1} \), and from Step 2, \( \sqrt{147}=7\sqrt{3} \) and \( \sqrt{-1}=i \). So \( \sqrt{-147}=7\sqrt{3}i \).

Answer:

\( 7i\sqrt{3} \)