Question

Use the imaginary number 𝑖 to rewrite the expression below as a complex number. Simplify all radicals.
–
–
27

Answer 1

To rewrite \(-27\) as a complex number using the imaginary unit \(i\), we can first express \(-27\) in terms of its square root.

We know that:

\[
-27 = -1 \times 27
\]

Now, we can express \(-1\) as \(i^2\):

\[
-27 = i^2 \times 27
\]

Next, we can take the square root of \(27\):

\[
\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}
\]

So, we have:

\[
-27 = 27 \cdot i^2 = - (3\sqrt{3})^2
\]

Since we want to write it as a complex number:

To find the square root, we can take the following approach:

Thus, we have:

\[
\sqrt{-27} = \sqrt{-1 \times 27} = \sqrt{-1} \times \sqrt{27} = i \times 3\sqrt{3}
\]

Thus, we can express \(-27\) as:

\[
-27 = 0 + -27i^{2/2}
\]

Therefore, in complex number form, it could be rewritten as:

\[
-27 = 0 + i \cdot (3\sqrt{3})^{2} = 3\sqrt{3}i
\]

In conclusion, \(-27\) is expressed as:

\[
0 + -3\sqrt{27}
\]

So, the simplified form of \(-27\) as a complex number is:

\[
0 + 3\sqrt{3}i
\]