To re-express the negative square root \(-\sqrt{-20}\) in the standard form \(a + bi\), we start by noting that:
\[ -\sqrt{-20} = -\sqrt{20} \cdot i \]
Next, we simplify \(\sqrt{20}\):
\[ \sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \]
Thus, we have:
\[ -\sqrt{-20} = -2\sqrt{5} \cdot i \]
This can be expressed in standard form as:
\[ 0 - 2\sqrt{5} i \]
So, the complex number is:
\[ 0 - 2\sqrt{5} i \]
Now, matching this to the options provided:
- \(-12i\sqrt{5}\) - This is not a match.
- \(-12\sqrt{5}i\) - This is a match as we found \(-2\sqrt{5} i\) which reflects a form of \(-12\sqrt{5} i\).
- \(-12\sqrt{5}\) - This is purely real, hence also not a match.
- \(2i\sqrt{5}\) - This is not a match since we have a negative sign.
None of the options directly match \(-2\sqrt{5} i\). However, interpreting two of the options more broadly can yield clearer results. The closest interpretation from your choices would be:
- \(-12i\sqrt{5}\) can sometimes be taken in context-related mathematical problems but does have a different constant.
Since these do not seem to perfectly align, please check the presented options again, as none of these representations match \(-2\sqrt{5} i\) directly. If there’s further simplification of notation that allows for a different interpretation, please provide that context again to navigate the right solution.
For clarity, the actual re-expressed form of \(-\sqrt{-20}\) is \(0 - 2\sqrt{5}i\), but none of the options appear to express that correctly.
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