To express the complex number [tex]\(\sqrt{-48} - 4\)[/tex] in the form [tex]\(a + bi\)[/tex], we need to simplify [tex]\(\sqrt{-48}\)[/tex] first. Here’s a step-by-step solution:
1. Simplify [tex]\(\sqrt{-48}\)[/tex]:
- We start by noting that [tex]\(\sqrt{-48}\)[/tex] can be separated into [tex]\(\sqrt{48} \cdot \sqrt{-1}\)[/tex] because [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex].
- We know [tex]\(\sqrt{-1} = i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit.
- Therefore, [tex]\(\sqrt{-48} = \sqrt{48} \cdot i\)[/tex].
2. Simplify [tex]\(\sqrt{48}\)[/tex]:
- [tex]\(\sqrt{48}\)[/tex] can be further simplified as [tex]\(\sqrt{16 \cdot 3}\)[/tex].
- Since [tex]\(\sqrt{16} = 4\)[/tex] and [tex]\(\sqrt{3}\)[/tex] remains as [tex]\(\sqrt{3}\)[/tex], we have [tex]\(\sqrt{48} = 4\sqrt{3}\)[/tex].
3. Combine the steps:
- Now, substituting back, we get [tex]\(\sqrt{-48} = 4\sqrt{3} \cdot i\)[/tex].
4. Subtract 4:
- We need to compute [tex]\(\sqrt{-48} - 4\)[/tex].
- Substituting [tex]\(\sqrt{-48}\)[/tex], we get [tex]\(4\sqrt{3}i - 4\)[/tex].
5. Express in the form [tex]\(a + bi\)[/tex]:
- The real part [tex]\(a\)[/tex] is [tex]\(-4\)[/tex].
- The imaginary part [tex]\(b\)[/tex] is [tex]\(4\sqrt{3}\)[/tex].
Thus, the expression [tex]\(\sqrt{-48} - 4\)[/tex] in the form [tex]\(a + bi\)[/tex] is [tex]\(-4 + 4\sqrt{3} i\)[/tex].
Comparing this with the given options:
- The correct answer is:
[tex]\[
-4 + 4i\sqrt{3}
\][/tex]
