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

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

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

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

[tex]\[ \square \][/tex]



Answer :

GinnyAnswer
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]

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