Question

Which of the following re-expresses the negative square root −6−20−−−−√ as a complex number in the standard form a+bi ?(1 point)
Responses

−125i−−√
negative 12 Start Root 5 i End Root

2i5–√
2 i Start Root 5 End Root

−12i5–√
negative 12 i Start Root 5 End Root

−125–√
negative 12 Start Root 5 End Root

Answer 1

To re-express the negative square root \(-6 - 20\sqrt{}\) as a complex number in standard form \(a + bi\), we first need to evaluate the expression inside the square root.

1. We start with the expression:
\[
-6 - 20
\]
Here, we can combine these terms:
\[
-6 - 20 = -26
\]

2. Now, we need to take the square root of \(-26\):
\[
\sqrt{-26} = \sqrt{26} \cdot \sqrt{-1} = \sqrt{26} i
\]

3. Since we want the negative square root:
\[
-\sqrt{-26} = -\sqrt{26} i
\]

4. We should write \(\sqrt{26}\) in terms of its prime factors. The prime factorization of 26 is \(2 \cdot 13\), but we can express this as:
\[
\sqrt{26} = \sqrt{2 \cdot 13} = \sqrt{2} \cdot \sqrt{13}
\]

However, for practical purposes in expressing as a complex number, we will leave it as \(\sqrt{26}\).

5. Finally, we can express the complex number in standard form:
\[
-\sqrt{26} i
\]

Now we compare this with the options provided.

The correct representation of \(-\sqrt{26} i\) in the options given is:
\[
-\sqrt{12} (\sqrt{5}) i = -12 \sqrt{5} i
\]
However, there might be some mix-up in the numbers in the options provided since \(-\sqrt{26} i\) doesn't match the given options perfectly.

Given this reasoning, find which expression fits. The likely intended option seems to be:
\[
-\sqrt{12 (\sqrt{5})} i = -12 \sqrt{5} i
\]
So, the most reasonable fit from the options is:
\(-12i \sqrt{5}\).

Check with your exam or task to ensure this reasoning aligns correctly, as none of the options given might perfectly match \(-\sqrt{26} i\).