To determine which expression is equivalent to [tex]\( \sqrt{-108} - \sqrt{-3} \)[/tex]:
First, let's simplify each term separately using imaginary numbers.
The square root of a negative number can be expressed using the imaginary unit [tex]\( i \)[/tex], where [tex]\( i = \sqrt{-1} \)[/tex].
1. Simplify [tex]\( \sqrt{-108} \)[/tex]:
[tex]\[
\sqrt{-108} = \sqrt{108} \cdot i
\][/tex]
Next, factorize 108 under the square root:
[tex]\[
108 = 36 \times 3
\][/tex]
So,
[tex]\[
\sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \cdot \sqrt{3} = 6\sqrt{3}
\][/tex]
Thus,
[tex]\[
\sqrt{-108} = 6\sqrt{3} \cdot i = 6i\sqrt{3}
\][/tex]
2. Simplify [tex]\( \sqrt{-3} \)[/tex]:
[tex]\[
\sqrt{-3} = \sqrt{3} \cdot i = i\sqrt{3}
\][/tex]
Now, we need to subtract these two results:
[tex]\[
\sqrt{-108} - \sqrt{-3} = 6i\sqrt{3} - i\sqrt{3}
\][/tex]
Factor out the common term [tex]\( i\sqrt{3} \)[/tex]:
[tex]\[
\sqrt{-108} - \sqrt{-3} = (6-1)i\sqrt{3} = 5i\sqrt{3}
\][/tex]
Therefore, the expression that is equivalent to [tex]\( \sqrt{-108} - \sqrt{-3} \)[/tex] is:
[tex]\[
\boxed{5i \sqrt{3}}
\][/tex]
