Look at the example below, which shows how the product property of radicals is used to simplify a radical:

[tex]\[
\sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i
\][/tex]

Use the product property of radicals to simplify the following radical:

[tex]\[
\sqrt{-81} = \square
\][/tex]



Answer :

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]

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