Radicals and rational expressions

Sure! Here’s a quick check on radicals and rational expressions. I’ll provide some sample questions and their solutions. You can try to solve them yourself first, and then look at the answers!

### Sample Questions:

1. **Simplify the radical:**
\(\sqrt{72}\)

2. **Simplify the expression:**
\(\frac{3x^4y^2}{6xy}\)

3. **Rationalize the denominator:**
\(\frac{5}{\sqrt{2}}\)

4. **Combine the rational expressions:**
\(\frac{1}{x} + \frac{2}{x^2}\)

5. **Simplify the expression:**
\(\frac{\sqrt{50}}{\sqrt{2}}\)

### Answers:

1. **Simplify the radical:**
\(\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}\)

2. **Simplify the expression:**
\(\frac{3x^4y^2}{6xy} = \frac{1}{2} x^{4-1}y^{2-1} = \frac{1}{2} x^3y\)

3. **Rationalize the denominator:**
\(\frac{5}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{5\sqrt{2}}{2}\)

4. **Combine the rational expressions:**
\(\frac{1}{x} + \frac{2}{x^2} = \frac{x + 2}{x^2}\)

5. **Simplify the expression:**
\(\frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}} = \sqrt{25} = 5\)

Feel free to attempt these problems, and let me know if you have any questions or need further clarification!