Which of the following re-expresses the negative square root −40−−−−√ as a complex number in the standard form a+bi ?(1 point) Responses 210−−√ 2 Start Root 10 End Root 2i10−−√ 2 i Start Root 10 End Root 4i10−−√ 4 i Start Root 10 End Root 210i−−−√

Answer 1

To express the negative square root $-\sqrt{40}$ as a complex number in standard form $a+bi$, we start by rewriting the square root:

1. We can simplify $-\sqrt{40}$: $$-\sqrt{40}=-\sqrt{4\cdot 10}=-\sqrt{4}\cdot \sqrt{10}=-2\sqrt{10}$$

2. 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:

1. $2\sqrt{10}2i$
2. $10\sqrt{2}i$
3. $4i\sqrt{10}$
4. $2i\sqrt{10}$

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.