To express the negative square root \(-\sqrt{40}\) as a complex number in standard form \(a + bi\), we start by rewriting the square root:
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We can simplify \(-\sqrt{40}\): \[ -\sqrt{40} = -\sqrt{4 \cdot 10} = -\sqrt{4} \cdot \sqrt{10} = -2\sqrt{10} \]
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Since we want to write this as a complex number in the form \(a + bi\), we note that a purely real negative number can also be expressed in terms of \(i\): \[ -2\sqrt{10} = 0 + (-2\sqrt{10})i \]
Thus, we can express \(-\sqrt{40}\) as: \[ 0 - 2\sqrt{10}i \] This is in the form \(a + bi\) where \(a = 0\) and \(b = -2\sqrt{10}\).
After reviewing the options you've provided:
- \(2\sqrt{10} 2i\)
- \(10\sqrt{2} i\)
- \(4i\sqrt{10}\)
- \(2i\sqrt{10}\)
We identify that none of the options directly express the negative square root \(-\sqrt{40}\) correctly.
However, if we only consider imaginary components, we see that the term with an \(i\) associated with \(\sqrt{10}\) is only present in the last option you provided:
- \(2i\sqrt{10}\)
However, since this option represents a positive value in terms of \(i\), it does not match with \(-2\sqrt{10}i\).
Therefore, based on your list, the expression \(-\sqrt{40}\) re-expressed correctly would be: \[ -2\sqrt{10}i \] but no option provided exactly matches this requirement.
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