To solve the expression [tex]\(\pm \sqrt{-27}\)[/tex], we need to remember that taking the square root of a negative number yields an imaginary number. Here's the step-by-step process to find [tex]\( \pm \sqrt{-27} \)[/tex]:
1. Identify the negative under the square root: We start with [tex]\(\sqrt{-27}\)[/tex].
2. Express in terms of imaginary numbers: Recall that [tex]\(\sqrt{-1} = i\)[/tex]. Therefore, we can rewrite the expression as:
[tex]\[
\sqrt{-27} = \sqrt{27 \cdot -1} = \sqrt{27} \cdot \sqrt{-1} = \sqrt{27} \cdot i
\][/tex]
3. Simplify the square root of the positive part: Next, we need to simplify [tex]\(\sqrt{27}\)[/tex]:
[tex]\[
\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}
\][/tex]
4. Combine the results: Putting these together, we have:
[tex]\[
\sqrt{-27} = 3\sqrt{3} \cdot i
\][/tex]
Therefore, the solutions are:
[tex]\[
+ \sqrt{-27} = +3\sqrt{3} \cdot i = 3\sqrt{3}i
\][/tex]
and
[tex]\[
- \sqrt{-27} = -3\sqrt{3} \cdot i = -3\sqrt{3}i
\][/tex]
5. Final answer: When expressed as a numerical form:
[tex]\[
3\sqrt{3}i \approx 5.196152422706632i
\][/tex]
Thus, the expression [tex]\(\pm \sqrt{-27} \)[/tex] results in:
[tex]\[
\pm \sqrt{-27} = \pm 5.196152422706632i
\][/tex]
Hence, the complete solution is:
[tex]\[
\pm \sqrt{-27} = 5.196152422706632i, -5.196152422706632i
\][/tex]
