Question

re-write the following: $3 + \sqrt{-4}$

• $-5i$
• $3 + 2i$
• $5i$
• $3 + i$

Explanation:

Step1: Recall the imaginary unit definition

The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \), so we can rewrite \( \sqrt{-4} \) by factoring out the negative sign.
\( \sqrt{-4} = \sqrt{4\times(-1)} \)

Step2: Use square root properties

Using the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a\geq0, b\geq0 \), and extended to complex numbers here), we have:
\( \sqrt{4\times(-1)}=\sqrt{4}\times\sqrt{-1} \)
Since \( \sqrt{4} = 2 \) and \( \sqrt{-1}=i \), this becomes \( 2i \).

Step3: Substitute back into the original expression

The original expression is \( 3+\sqrt{-4} \), substituting \( \sqrt{-4}=2i \) gives \( 3 + 2i \).

Answer:

B. \( 3 + 2i \)