To express \(-\sqrt{-20}\) as a complex number in standard form \(a + bi\), we can break it down as follows:
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Recognize that \(-\sqrt{-20}\) can be rewritten using imaginary numbers. We can factor out the negative inside the square root: \[ -\sqrt{-20} = -\sqrt{20} \cdot \sqrt{-1} = -\sqrt{20} \cdot i \]
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Next, simplify \(\sqrt{20}\): \[ \sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \]
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Substitute this back into the expression: \[ -\sqrt{-20} = -2\sqrt{5} \cdot i \]
Thus, \(-\sqrt{-20}\) can be expressed in standard form \(a + bi\) as: \[ 0 - 2\sqrt{5}i \]
In terms of the components:
- \(a = 0\)
- \(b = -2\sqrt{5}\)
So the expression in standard form is: \[ 0 - 2\sqrt{5} i \]
Among your options, it looks like the equivalent form can be represented as: \(-2\sqrt{5} i\)
Therefore, the correct response from your provided options would be: \(-12 i \sqrt{5}\) (assuming a typo or mix-up in the presentation). If indeed \( -2\sqrt{5}i\) was present discretely as an option, you also may find \( -12i \sqrt{5} \) as \( -12\sqrt{5} i \). From the choices you've shared, the most likely intended answer is: \(-2i\sqrt{5}\).
However, please double-check the options presented to ensure they align correctly with the above derivation.
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