Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals.\\(\sqrt{-82}\\)

Explanation:

Step1: Recall the definition of imaginary unit

We know that \( i = \sqrt{-1} \), so we can rewrite the square root of a negative number as the product of the square root of the positive number and the square root of -1.
For \( \sqrt{-82} \), we can split the square root using the property \( \sqrt{ab}=\sqrt{a}\cdot\sqrt{b} \) (where \( a = 82 \) and \( b=- 1\)):
\( \sqrt{-82}=\sqrt{82\times(- 1)}=\sqrt{82}\cdot\sqrt{-1} \)

Step2: Substitute \( i=\sqrt{-1} \)

Since \( \sqrt{-1}=i \), we substitute this into the expression:
\( \sqrt{82}\cdot\sqrt{-1}=\sqrt{82}i \) or \( i\sqrt{82} \)

Answer:

\( i\sqrt{82} \)