Question

look at the example below which shows how the product property of radicals is used to simplify a radical.
√(-36) = √36√(-1) = 6i
use the product property of radicals to simplify the following radical.
√(-81) =
done

Explanation:

Step1: Apply product property of radicals

The product property of radicals states that \(\sqrt{ab}=\sqrt{a}\sqrt{b}\) (for appropriate \(a,b\)). For \(\sqrt{-81}\), we can write \(-81 = 81\times(-1)\), so \(\sqrt{-81}=\sqrt{81\times(-1)}\). By the product property, this is \(\sqrt{81}\sqrt{-1}\).

Step2: Simplify square roots and \(i\)

We know that \(\sqrt{81} = 9\) (since \(9\times9 = 81\)) and \(\sqrt{-1}=i\) (by definition of the imaginary unit \(i\)). So substituting these values in, we get \(9\times i=9i\).

Answer:

\(9i\)