Question

- 1/3 square root-90

Answer 1

Answer: $-i\sqrt{10}$

Step-by-step explanation:

Assuming that you need to simplify the expression, below is the explanation to do it.

Given the following expression:

$-\frac{1}{3}\sqrt{-90}$

You need to decompose the radicand (The number inside the square root) into its prime factors:

$90=2*3*3*5=2*3^2*5$

Knowing that, you can rewrite the expression in this form:

$=-\frac{(1)(\sqrt{-2*3^2*5})}{3}=-\frac{\sqrt{-2*3^2*5}}{3}$

Since $\sqrt{-1}=i$, you must substitute it into the expression:

$=-\frac{i\sqrt{2*3^2*5}}{3}$

Now you need to remember the following property:

$\sqrt[n]{a^n}=a^{\frac{n}{n}}=a$

Then, applying that property, you get:i:

$=-\frac{3i\sqrt{2*5}}{3}=-\frac{3i\sqrt{10}}{3}$

Finally, you must divide the numerator and the denominator by 3. So, you get:

$=-\frac{i\sqrt{10}}{1}=-i\sqrt{10}$