divide. (\frac{sqrt{-75}}{sqrt{-27}}) (simplify your answer. type an exact answer, using radicals as needed…

Explanation:

Step1: Rewrite square - roots of negative numbers

Recall that $\sqrt{-a}=i\sqrt{a}$ for $a>0$. So, $\sqrt{-75}=i\sqrt{75}$ and $\sqrt{-27}=i\sqrt{27}$. Then $\frac{\sqrt{-75}}{\sqrt{-27}}=\frac{i\sqrt{75}}{i\sqrt{27}}$.

Step2: Simplify the fraction

Cancel out the $i$ terms. We get $\frac{\sqrt{75}}{\sqrt{27}}$. Using the property $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$ ($a\geq0,b > 0$), we have $\sqrt{\frac{75}{27}}$.

Step3: Simplify the fraction inside the square - root

Reduce $\frac{75}{27}$ to $\frac{25}{9}$ by dividing both the numerator and denominator by 3. So, $\sqrt{\frac{25}{9}}$.

Step4: Evaluate the square - root

Since $\sqrt{\frac{25}{9}}=\frac{\sqrt{25}}{\sqrt{9}}$, and $\sqrt{25} = 5$, $\sqrt{9}=3$, the result is $\frac{5}{3}$.

Answer:

$\frac{5}{3}$