To solve the expression [tex]\(\sqrt{-108} - \sqrt{-3}\)[/tex], let's break it down step by step.
1. Simplify [tex]\(\sqrt{-108}\)[/tex]:
- We know that [tex]\(\sqrt{-1} = i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit.
- Thus, [tex]\(\sqrt{-108}\)[/tex] can be expressed as [tex]\(\sqrt{108} \cdot i\)[/tex].
Next, we simplify [tex]\(\sqrt{108}\)[/tex]:
[tex]\[ \sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \cdot \sqrt{3} = 6\sqrt{3} \][/tex]
Therefore,
[tex]\[ \sqrt{-108} = 6\sqrt{3} \cdot i = 6\sqrt{3}i \][/tex]
2. Simplify [tex]\(\sqrt{-3}\)[/tex]:
- Similarly, we express [tex]\(\sqrt{-3}\)[/tex] as [tex]\(\sqrt{3} \cdot i\)[/tex].
[tex]\[ \sqrt{-3} = \sqrt{3} \cdot i = \sqrt{3}i \][/tex]
3. Subtract the simplified expressions:
[tex]\[ \sqrt{-108} - \sqrt{-3} = 6\sqrt{3}i - \sqrt{3}i \][/tex]
4. Combine like terms:
Factor out the common term [tex]\(\sqrt{3}i\)[/tex]:
[tex]\[ 6\sqrt{3}i - \sqrt{3}i = (6 - 1)\sqrt{3}i = 5\sqrt{3}i \][/tex]
Hence, the expression equivalent to [tex]\(\sqrt{-108} - \sqrt{-3}\)[/tex] is:
[tex]\[ 5 i \sqrt{3} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{5i \sqrt{3}} \][/tex]
