Simplify ( square root of -500)/( square root of -5)

Solution:

Extract the imaginary unit $i$.

Step 1: Express $-500$ as $-1\times 500$.

$\frac{\sqrt{-1\times 500}}{\sqrt{-5}}$

Step 2: Separate the square root of $-1$ and $500$.

$\frac{\sqrt{-1}\cdot \sqrt{500}}{\sqrt{-5}}$

Step 3: Replace $\sqrt{-1}$ with $i$.

$\frac{i\cdot \sqrt{500}}{\sqrt{-5}}$

Step 4: Express $-5$ as $-1\times 5$.

$\frac{i\cdot \sqrt{500}}{\sqrt{-1\times 5}}$

Step 5: Separate the square root of $-1$ and $5$.

$\frac{i\cdot \sqrt{500}}{\sqrt{-1}\cdot \sqrt{5}}$

Step 6: Replace $\sqrt{-1}$ with $i$ again.

$\frac{i\cdot \sqrt{500}}{i\cdot \sqrt{5}}$

Eliminate the common imaginary unit $i$.

Step 7: Remove the $i$ from numerator and denominator.

$\frac{\cancel{i}\cdot \sqrt{500}}{\cancel{i}\cdot \sqrt{5}}$

Step 8: Simplify the expression.

$\frac{\sqrt{500}}{\sqrt{5}}$

Combine the square roots.

Step 9: Merge the square roots into one.

$\sqrt{\frac{500}{5}}$

Simplify the radical expression.

Step 10: Calculate $\frac{500}{5}$.

$\sqrt{100}$

Step 11: Recognize $100$ as a perfect square, $(10{)}^2$.

$\sqrt{(10{)}^2}$

Step 12: Simplify to obtain the final result.

$10$

Knowledge Notes:

To solve the given problem, we utilize several mathematical principles and properties:

1. Complex Numbers and Imaginary Unit ($i$): The imaginary unit $i$ is defined as $\sqrt{-1}$. It's used to express the square root of a negative number, which is not possible within the set of real numbers.

2. Properties of Square Roots: The square root of a product can be expressed as the product of the square roots of the individual factors, i.e., $\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}$.

3. Simplifying Square Roots: When simplifying square roots, we look for factors that are perfect squares, as they can be taken out of the square root as their square root value.

4. Rationalizing the Denominator: When we have a complex number in the denominator, we can eliminate it by multiplying both the numerator and the denominator by the complex conjugate of the denominator. In this case, however, we simply cancel out the common $i$ in both the numerator and the denominator.

5. Simplification: The final step involves simplifying the expression by performing arithmetic operations and recognizing perfect squares to extract them from the square root.

By following these steps, we can simplify the given expression involving complex numbers and square roots.