To simplify the radical expression [tex]\(\frac{3-\sqrt{-72}}{-6}\)[/tex], follow these steps:
1. Identify and handle the imaginary unit [tex]\(i\)[/tex]:
Recall that [tex]\(\sqrt{-72}\)[/tex] can be written using the imaginary unit [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex]. Thus, [tex]\(\sqrt{-72} = \sqrt{72} \cdot i\)[/tex].
2. Simplify the square root:
Notice that [tex]\(\sqrt{72}\)[/tex] can be simplified. Since [tex]\(72 = 36 \cdot 2\)[/tex], we have [tex]\(\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}\)[/tex].
Therefore, [tex]\(\sqrt{-72} = 6\sqrt{2} \cdot i\)[/tex].
3. Substitute back into the original expression:
Replacing [tex]\(\sqrt{-72}\)[/tex] with [tex]\(6\sqrt{2} \cdot i\)[/tex], the expression becomes:
[tex]\[
\frac{3 - 6\sqrt{2} \cdot i}{-6}
\][/tex]
4. Separate the real and imaginary parts:
Break down the fraction to handle the numerator and the denominator separately:
[tex]\[
\frac{3}{-6} - \frac{6\sqrt{2} \cdot i}{-6}
\][/tex]
5. Simplify each term individually:
For the real part:
[tex]\[
\frac{3}{-6} = -\frac{1}{2}
\][/tex]
For the imaginary part:
[tex]\[
\frac{6\sqrt{2} \cdot i}{-6} = -\sqrt{2} \cdot i
\][/tex]
6. Combine the simplified real and imaginary parts:
Putting it all together, we have:
[tex]\[
-\frac{1}{2} + \sqrt{2} \cdot i
\][/tex]
Thus, the simplified radical expression is:
[tex]\[
\boxed{-\frac{1}{2} + \sqrt{2} \cdot i}
\][/tex]
