To simplify [tex]\(\sqrt{-81}\)[/tex] using the product property of radicals, let's follow these steps:
1. Identify the Problem:
[tex]\[
\sqrt{-81}
\][/tex]
2. Apply the Product Property of Radicals:
The product property of radicals states that the square root of a product is equal to the product of the square roots of the factors. Therefore:
[tex]\[
\sqrt{-81} = \sqrt{81} \times \sqrt{-1}
\][/tex]
3. Simplify Each Factor:
- [tex]\(\sqrt{81}\)[/tex] is a perfect square.
[tex]\[
\sqrt{81} = 9
\][/tex]
- [tex]\(\sqrt{-1}\)[/tex] is defined as the imaginary unit [tex]\(i\)[/tex].
[tex]\[
\sqrt{-1} = i
\][/tex]
4. Combine the Results:
Multiplying the simplified components together:
[tex]\[
\sqrt{-81} = 9 \times i
\][/tex]
5. Write the Final Answer:
[tex]\[
\sqrt{-81} = 9i
\][/tex]
So, by using the product property of radicals, we have simplified [tex]\(\sqrt{-81}\)[/tex] to:
[tex]\[
\boxed{9i}
\][/tex]
