To simplify the expression [tex]\(3 \sqrt{-75}\)[/tex], follow these steps:
1. Factorize the Number Inside the Square Root:
[tex]\[
-75 = -1 \times 75 = -1 \times 25 \times 3
\][/tex]
2. Apply the Property of Square Roots:
The property [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex] allows us to break down the square root:
[tex]\[
\sqrt{-75} = \sqrt{-1 \times 25 \times 3} = \sqrt{-1} \times \sqrt{25} \times \sqrt{3}
\][/tex]
3. Simplify Each Part of the Expression:
- [tex]\(\sqrt{-1} = i\)[/tex] (where [tex]\(i\)[/tex] is the imaginary unit),
- [tex]\(\sqrt{25} = 5\)[/tex].
Combining these results, we get:
[tex]\[
\sqrt{-75} = i \times 5 \times \sqrt{3} = 5i \sqrt{3}
\][/tex]
4. Multiply by 3:
Now, multiply the simplified square root expression by 3:
[tex]\[
3 \sqrt{-75} = 3 \times 5i \sqrt{3} = 15i \sqrt{3}
\][/tex]
Thus, the expression equal to [tex]\(3 \sqrt{-75}\)[/tex] is:
[tex]\[
15 i \sqrt{3}
\][/tex]
Therefore, the correct answer is:
[tex]$15 i \sqrt{3}$[/tex].
