To simplify the radical [tex]\(\sqrt{-81}\)[/tex] using the product property of radicals, follow these steps:
1. Identify the radicand and decompose it:
[tex]\[
\sqrt{-81}
\][/tex]
2. Use the product property of radicals:
The product property of radicals tells us that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]. In this case, we can write:
[tex]\[
\sqrt{-81} = \sqrt{81 \cdot -1}
\][/tex]
3. Break the radicand into separate square roots:
We can now separate [tex]\(\sqrt{81 \cdot -1}\)[/tex] into [tex]\(\sqrt{81}\)[/tex] and [tex]\(\sqrt{-1}\)[/tex]:
[tex]\[
\sqrt{-81} = \sqrt{81} \cdot \sqrt{-1}
\][/tex]
4. Simplify each square root individually:
- The square root of [tex]\(81\)[/tex] is [tex]\(9\)[/tex]:
[tex]\[
\sqrt{81} = 9
\][/tex]
- The square root of [tex]\(-1\)[/tex] is represented by the imaginary unit [tex]\(i\)[/tex]:
[tex]\[
\sqrt{-1} = i
\][/tex]
5. Combine the results:
Now, multiply the simplified components together:
[tex]\[
\sqrt{81} \cdot \sqrt{-1} = 9 \cdot i
\][/tex]
Therefore, the simplified form of the radical [tex]\(\sqrt{-81}\)[/tex] is:
[tex]\[
\sqrt{-81} = 9i
\][/tex]
