Question

1. write \\(\sqrt{-48} - 4\\) in the form of \\(a + bi\\).

\\(-4 - 4i\sqrt{3}\\)

\\(-4 + 48i\\)

\\(-4 + 4i\sqrt{3}\\)

\\(-4 - i\sqrt{48}\\)

Explanation:

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<reasoning>

Identify the components of the expression

Using the Complex Numbers and Imaginary Numbers knowledge points
We are given the expression \(\sqrt{-48} - 4\) and need to rewrite it in standard form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

Simplify the radical term

Using the Imaginary Numbers knowledge point
\[
\sqrt{-48} = \sqrt{-1 \cdot 16 \cdot 3} = \sqrt{-1} \cdot \sqrt{16} \cdot \sqrt{3} = i \cdot 4 \cdot \sqrt{3} = 4i\sqrt{3}
\]

Rearrange into standard form

Using the Complex Numbers knowledge point
\[
\sqrt{-48} - 4 = 4i\sqrt{3} - 4 = -4 + 4i\sqrt{3}
\]
</reasoning>

<answer>
<mcq-option>(A) \(-4 - 4i\sqrt{3}\)</mcq-option>
<mcq-option>(B) \(-4 + 48i\)</mcq-option>
<mcq-correct>(C) \(-4 + 4i\sqrt{3}\)</mcq-correct>
<mcq-option>(D) \(-4 - i\sqrt{48}\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the components of the expression

Using the Complex Numbers and Imaginary Numbers knowledge points
We are given the expression \(\sqrt{-48} - 4\) and need to rewrite it in standard form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

Simplify the radical term

Using the Imaginary Numbers knowledge point
\[
\sqrt{-48} = \sqrt{-1 \cdot 16 \cdot 3} = \sqrt{-1} \cdot \sqrt{16} \cdot \sqrt{3} = i \cdot 4 \cdot \sqrt{3} = 4i\sqrt{3}
\]

Rearrange into standard form

Using the Complex Numbers knowledge point
\[
\sqrt{-48} - 4 = 4i\sqrt{3} - 4 = -4 + 4i\sqrt{3}
\]
</reasoning>

<answer>
<mcq-option>(A) \(-4 - 4i\sqrt{3}\)</mcq-option>
<mcq-option>(B) \(-4 + 48i\)</mcq-option>
<mcq-correct>(C) \(-4 + 4i\sqrt{3}\)</mcq-correct>
<mcq-option>(D) \(-4 - i\sqrt{48}\)</mcq-option>
</answer>

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