To determine which expression is equivalent to [tex]\(\sqrt{-108} - \sqrt{-3}\)[/tex], let's go through the problem step by step.
Firstly, we need to remember that the square root of a negative number can be expressed in terms of imaginary units, where [tex]\(i = \sqrt{-1}\)[/tex].
1. Calculate [tex]\(\sqrt{-108}\)[/tex]:
We rewrite [tex]\(\sqrt{-108}\)[/tex] as [tex]\(\sqrt{108} \cdot \sqrt{-1} = \sqrt{108} \cdot i\)[/tex].
- Simplifying [tex]\(\sqrt{108}\)[/tex]:
[tex]\(\sqrt{108}\)[/tex] can be simplified as [tex]\(\sqrt{36 \cdot 3} = \sqrt{36} \cdot \sqrt{3} = 6 \sqrt{3}\)[/tex].
Thus,
[tex]\[\sqrt{-108} = 6 \sqrt{3} \cdot i = 6i \sqrt{3}.\][/tex]
2. Calculate [tex]\(\sqrt{-3}\)[/tex]:
Similarly, rewrite [tex]\(\sqrt{-3}\)[/tex] as [tex]\(\sqrt{3} \cdot \sqrt{-1} = \sqrt{3} \cdot i\)[/tex].
- This means:
[tex]\[\sqrt{-3} = \sqrt{3} \cdot i = i \sqrt{3}.\][/tex]
3. Subtract the expressions:
Now, subtract [tex]\(\sqrt{-3}\)[/tex] from [tex]\(\sqrt{-108}\)[/tex]:
[tex]\[
\sqrt{-108} - \sqrt{-3} = 6i \sqrt{3} - i \sqrt{3}.
\][/tex]
4. Factor out the common term [tex]\(i \sqrt{3}\)[/tex]:
[tex]\[
6i \sqrt{3} - i \sqrt{3} = (6 - 1) i \sqrt{3} = 5i \sqrt{3}.
\][/tex]
Therefore, the expression equivalent to [tex]\(\sqrt{-108} - \sqrt{-3}\)[/tex] is:
[tex]\[
\boxed{5i \sqrt{3}}.
\][/tex]
