Questions: Divide. [ fracsqrt-57sqrt3 fracsqrt-57sqrt3= ] (Simplify your answer. Type

Solution Steps

To divide the square roots of negative numbers, we can use the property of square roots and imaginary numbers. Specifically, we can express the square roots of negative numbers in terms of the imaginary unit \(i\), where \(i = \sqrt{-1}\). Then, simplify the expression by dividing the coefficients and simplifying the square roots.

We start with the expression

\[ \frac{\sqrt{-57}}{\sqrt{3}}. \]

Using the property of imaginary numbers, we can rewrite the square root of a negative number as

\[ \sqrt{-57} = \sqrt{57} \cdot i. \]

Thus, we have

\[ \frac{\sqrt{-57}}{\sqrt{3}} = \frac{\sqrt{57} \cdot i}{\sqrt{3}}. \]

Next, we can simplify the expression by separating the imaginary unit and the square roots:

\[ \frac{\sqrt{57}}{\sqrt{3}} \cdot i = \sqrt{\frac{57}{3}} \cdot i = \sqrt{19} \cdot i. \]

Calculating the numerical value of \(\sqrt{19}\) gives approximately \(4.3589\). Therefore, we can express the result as:

\[ \frac{\sqrt{-57}}{\sqrt{3}} \approx 4.3589i. \]

Final Answer