Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals.
$-sqrt{-21}$

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 \( \sqrt{a}\cdot i \).
For the expression \( -\sqrt{-21} \), we can split the square root of the negative number:
\( -\sqrt{-21}=-\sqrt{21\times(-1)} \)

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 \( b = - 1\) with the definition of \( i \)), we have:
\( -\sqrt{21\times(-1)}=-\sqrt{21}\cdot\sqrt{-1} \)

Step3: Substitute \( i=\sqrt{-1} \)

Since \( \sqrt{-1}=i \), we substitute this into the expression:
\( -\sqrt{21}\cdot\sqrt{-1}=-i\sqrt{21} \)

Answer:

\( -i\sqrt{21} \)