Question

imaginary unit
consider that ( i = sqrt{-1} ).
use the imaginary number ( i ) to rewrite the expression ( sqrt{-81} )
a ( -81i )
b ( -9 )
c ( -9i )
d ( 9i )

Explanation:

Step1: Rewrite the radicand

We know that \(\sqrt{-81}=\sqrt{81\times(-1)}\). By the property of square roots \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\), here we extend it to complex numbers), we can split the square root: \(\sqrt{81\times(-1)}=\sqrt{81}\times\sqrt{-1}\).

Step2: Simplify each square root

We know that \(\sqrt{81} = 9\) and \(\sqrt{-1}=i\) (by the definition of the imaginary unit \(i\)). So substituting these values in, we get \(\sqrt{81}\times\sqrt{-1}=9\times i = 9i\).

Answer:

D. \(9i\)