To solve [tex]\(6 \sqrt{-\frac{1}{9}}\)[/tex], let's go through the solution step-by-step:
1. Identify the expression inside the square root:
The given expression is [tex]\(6 \sqrt{-\frac{1}{9}}\)[/tex].
2. Understand the nature of the square root of a negative number:
The square root of a negative number involves imaginary numbers. Specifically, [tex]\(\sqrt{-1} = i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit.
3. Separate the square root into real and imaginary parts:
Rewrite the expression inside the square root:
[tex]\[
\sqrt{-\frac{1}{9}} = \sqrt{-1 \cdot \frac{1}{9}}
\][/tex]
4. Simplify the term inside the square root:
You can split this into two separate square roots:
[tex]\[
\sqrt{-1 \cdot \frac{1}{9}} = \sqrt{-1} \cdot \sqrt{\frac{1}{9}}
\][/tex]
5. Calculate [tex]\(\sqrt{-1}\)[/tex] and [tex]\(\sqrt{\frac{1}{9}}\)[/tex]:
[tex]\[
\sqrt{-1} = i
\][/tex]
[tex]\[
\sqrt{\frac{1}{9}} = \frac{1}{3}
\][/tex]
6. Multiply the results:
[tex]\[
\sqrt{-\frac{1}{9}} = i \cdot \frac{1}{3} = \frac{i}{3}
\][/tex]
7. Multiply this by 6:
[tex]\[
6 \sqrt{-\frac{1}{9}} = 6 \cdot \frac{i}{3}
\][/tex]
8. Simplify the multiplication:
[tex]\[
6 \cdot \frac{i}{3} = 2i
\][/tex]
Therefore, the value of [tex]\(6 \sqrt{-\frac{1}{9}}\)[/tex] is [tex]\(\boxed{2i}\)[/tex].
