Question

question
write \\(\sqrt{-28}\\) in simplest radical form.

Explanation:

Step1: Recall imaginary unit

The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \), so we can rewrite \( \sqrt{-28} \) as \( \sqrt{28 \times (-1)} \).

Step2: Use square - root property

Using the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a\geq0,b\geq0 \), and we extend it to the case where \( b = - 1 \) with the help of the imaginary unit), we have \( \sqrt{28\times(-1)}=\sqrt{28}\times\sqrt{-1} \). Since \( \sqrt{-1}=i \), we get \( i\sqrt{28} \).

Step3: Simplify \( \sqrt{28} \)

Factor 28: \( 28 = 4\times7 \). Then \( \sqrt{28}=\sqrt{4\times7} \). Using the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) again (where \( a = 4,b = 7 \) and \( a\geq0,b\geq0 \)), we have \( \sqrt{4\times7}=\sqrt{4}\times\sqrt{7} \). Since \( \sqrt{4} = 2 \), we get \( 2\sqrt{7} \).

Step4: Combine the results

Substitute \( \sqrt{28}=2\sqrt{7} \) back into \( i\sqrt{28} \), we get \( 2i\sqrt{7} \) or \( 2\sqrt{7}i \).

Answer:

\( 2i\sqrt{7} \)