To find an expression equivalent to \(\sqrt{-108} - \sqrt{-3}\), let's go through the simplification step-by-step.
1. First, understand that the square root of a negative number involves the imaginary unit \(i\), where \(i = \sqrt{-1}\).
2. Begin with \(\sqrt{-108}\):
- Rewrite \(\sqrt{-108}\) as \(\sqrt{108} \cdot \sqrt{-1}\).
- The \(\sqrt{-1}\) is \(i\), so this becomes \(\sqrt{108} \cdot i\).
- Simplify \(\sqrt{108}\): \(\sqrt{108} = \sqrt{36 \cdot 3} = \sqrt{36} \cdot \sqrt{3} = 6\sqrt{3}\).
- Therefore, \(\sqrt{-108} = 6\sqrt{3} \cdot i\) or \(6i\sqrt{3}\).
3. Now, for \(\sqrt{-3}\):
- Rewrite \(\sqrt{-3}\) as \(\sqrt{3} \cdot \sqrt{-1}\).
- The \(\sqrt{-1}\) is \(i\), so this becomes \(\sqrt{3} \cdot i\).
- Therefore, \(\sqrt{-3} = i\sqrt{3}\).
4. Subtract these two terms:
[tex]\[
\sqrt{-108} - \sqrt{-3} = 6i\sqrt{3} - i\sqrt{3}.
\][/tex]
- Factor out \(i\sqrt{3}\):
[tex]\[
(6i\sqrt{3}) - (i\sqrt{3}) = i\sqrt{3}(6 - 1) = 5i\sqrt{3}.
\][/tex]
Thus, the expression equivalent to \(\sqrt{-108} - \sqrt{-3}\) is \(5i\sqrt{3}\).
So, the answer is:
5i [tex]\(\sqrt{3}\)[/tex]
