Answer: [tex]-i\sqrt{10}[/tex]
Step-by-step explanation:
Assuming that you need to simplify the expression, below is the explanation to do it.
Given the following expression:
[tex]-\frac{1}{3}\sqrt{-90}[/tex]
You need to decompose the radicand (The number inside the square root) into its prime factors:
[tex]90=2*3*3*5=2*3^2*5[/tex]
Knowing that, you can rewrite the expression in this form:
[tex]=-\frac{(1)(\sqrt{-2*3^2*5})}{3}=-\frac{\sqrt{-2*3^2*5}}{3}[/tex]
Since [tex]\sqrt{-1}=i[/tex], you must substitute it into the expression:
[tex]=-\frac{i\sqrt{2*3^2*5}}{3}[/tex]
Now you need to remember the following property:
[tex]\sqrt[n]{a^n}=a^{\frac{n}{n}}=a[/tex]
Then, applying that property, you get:i:
[tex]=-\frac{3i\sqrt{2*5}}{3}=-\frac{3i\sqrt{10}}{3}[/tex]
Finally, you must divide the numerator and the denominator by 3. So, you get:
[tex]=-\frac{i\sqrt{10}}{1}=-i\sqrt{10}[/tex]
