Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals. \\(sqrt{-81}\\)

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 \( \sqrt{-81} \), we can split the square root: \( \sqrt{-81}=\sqrt{81\times(-1)} \)

Step2: Use 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 complex numbers), we have \( \sqrt{81\times(-1)}=\sqrt{81}\cdot\sqrt{-1} \)
Since \( \sqrt{81} = 9 \) and \( \sqrt{-1}=i \), then \( \sqrt{81}\cdot\sqrt{-1}=9i \)

Answer:

\( 9i \)