Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals.\\(\sqrt{-38}\\)

Explanation:

Step1: Recall the definition of imaginary unit

The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \), so we can rewrite \( \sqrt{-a} \) (where \( a>0 \)) as \( \sqrt{a}\cdot\sqrt{-1} \).
For the expression \( \sqrt{-38} \), we can split the square root: \( \sqrt{-38}=\sqrt{38\times(-1)} \)

Step2: Apply the property of square roots

Using the property \( \sqrt{ab}=\sqrt{a}\cdot\sqrt{b} \) (for \( a\geq0, b\geq0 \); here we extend it to the case where \( b = - 1 \) with the definition of \( i \)), we have \( \sqrt{38\times(-1)}=\sqrt{38}\cdot\sqrt{-1} \)
Since \( \sqrt{-1}=i \), then \( \sqrt{38}\cdot\sqrt{-1}=i\sqrt{38} \)

Answer:

\( i\sqrt{38} \)