Question

Re-express −18−−−−√ as a complex number with the imaginary unit i=−1−−−√ .

Answer 1

To re-express \(-18\) as a complex number using the imaginary unit \(i\), we start by recognizing that the imaginary unit is defined as \(i = \sqrt{-1}\).

To express \(-18\) as a complex number, we can write:

\[
-18 = -1 \cdot 18
\]

Since \(-1\) can be represented as \(i^2\), we have:

\[
-18 = 18 \cdot (-1) = 18 \cdot i^2
\]

Next, we can express this in terms of the square root:

\[
-18 = 18 \cdot \sqrt{-1}^2 = 18 \cdot (i)^2
\]

This means we can rewrite \(-18\) inside the square root:

\[
-18 = \sqrt{(-18)^2} = \sqrt{(-1) \cdot 18^2} = \sqrt{-1} \cdot \sqrt{18^2} = i \cdot 18
\]

Now, since \(\sqrt{-18} = \sqrt{-1} \cdot \sqrt{18} = i \sqrt{18}\), we can simplify \(\sqrt{18}\):

\[
\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}
\]

So now we can express \(-18\) in terms of \(i\):

\[
\sqrt{-18} = i \cdot 3\sqrt{2}
\]

Thus, we can conclude that:

\[
\sqrt{-18} = 3\sqrt{2} i
\]

In conclusion:

\[
-18 = \sqrt{-18} = 3\sqrt{2} i
\]

Therefore, \(-18\) can be expressed as:

\[
-18 = 0 + (-18)i
\]

Or simply,

\[
-18 = 0 + 3\sqrt{2} i
\]