Rationalize the denominator. Assume all variables represent positive real numbers. Simplify your answer and type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression: -sqrt(8r^3/s)

To rationalize the denominator and simplify the expression \(-\sqrt{\frac{8r^3}{s}}\), follow these steps:

1. **Rewrite the expression**:
   \[
   -\sqrt{\frac{8r^3}{s}} = -\frac{\sqrt{8r^3}}{\sqrt{s}}
   \]

2. **Simplify the square roots**:
   - The square root of \(8r^3\) can be simplified as follows:
     \[
     \sqrt{8r^3} = \sqrt{4 \times 2 \times r^2 \times r} = \sqrt{4r^2 \times 2r} = 2r\sqrt{2r}
     \]
   - The expression now becomes:
     \[
     -\frac{2r\sqrt{2r}}{\sqrt{s}}
     \]

3. **Rationalize the denominator** by multiplying the numerator and the denominator by \(\sqrt{s}\):
   \[
   -\frac{2r\sqrt{2r}}{\sqrt{s}} \times \frac{\sqrt{s}}{\sqrt{s}} = -\frac{2r\sqrt{2rs}}{s}
   \]

4. **Write the final simplified expression**:
   \[
   -\frac{2r\sqrt{2rs}}{s}
   \]

Thus, the rationalized and simplified form of the expression is \( -\frac{2r\sqrt{2rs}}{s} \).

Would you like more details or have any questions?

Here are 8 related questions you might find helpful:
1. How do you rationalize the denominator when the denominator is a binomial?
2. Can you explain why rationalizing the denominator is important?
3. How do you simplify expressions involving higher roots?
4. What happens if the expression contains complex numbers instead of real numbers?
5. How do you handle rationalizing when the denominator is a sum of radicals?
6. What are the general steps for simplifying radical expressions?
7. Can you rationalize a denominator that is already a rational number?
8. How does the process differ when dealing with cube roots or higher roots?

**Tip:** When rationalizing the denominator, remember to multiply by a form of 1 that includes the radical in the denominator to eliminate it.

Learn how to rationalize the denominator and simplify \(-\sqrt{\frac{8r^3}{s}}\) step-by-step with detailed explanations. This problem involves simplifying radicals and rationalizing denominators in algebra. Suitable for high school students studying algebra.

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