Answer
$\sqrt \frac{1}{3}\sqrt -27=3i$
Work Step by Step
Since $i=\sqrt -1$ we can write $\sqrt \frac{1}{3}\times\sqrt -27=\sqrt \frac{1}{3}\times i\sqrt 27$.
We can write $i\sqrt 27=i\sqrt (3\times9)=3i\sqrt 3$ hence $\sqrt \frac{1}{3}\times i\sqrt 27=\sqrt \frac{1}{3}\times3i\sqrt 3=\frac{1}{\sqrt 3}\times3i\sqrt 3 =\frac{3i\sqrt 3}{\sqrt 3}$. The $\sqrt 3$'s then cancel each other which leaves a final answer of $3i$.
Hence the final answer written in the form $a+bi$ where $a$ is the real part and $b$ is the imaginary part is $3i$ as $a=0$ and $b=3$.
