Operations with Radicals

Operations with Radicals

Simplify each radical expression. Perform the indicated operations and rationalize denominators where necessary. Show all your work.

1. Simplify the following radical expression: \(\sqrt{72x^5y^8}\)

2. Simplify: \(\sqrt[3]{108a^7b^{10}}\)

3. Perform the operation: \(3\sqrt{20} - 2\sqrt{45} + \sqrt{80}\)

4. Simplify: \(5\sqrt[3]{16} + \sqrt[3]{54} - 2\sqrt[3]{250}\)

5. Multiply and simplify: \((2\sqrt{3} + 5)(4\sqrt{3} - 1)\)

6. Expand and simplify: \((3\sqrt{x} - \sqrt{y})^2\)

7. Rationalize the denominator and simplify: \(\frac{6}{\sqrt{3}}\)

8. Rationalize the denominator and simplify: \(\frac{5}{2 - \sqrt{3}}\)

9. Rationalize the denominator and simplify: \(\frac{\sqrt{x} + \sqrt{y}}{\sqrt{x} - \sqrt{y}}\)