Problems

1. Simplify. Assume that the variables represent positive real numbers.

   (a). $\sqrt{10y^3}\sqrt{5y}$

   (b). $\sqrt{2b^8}\sqrt{18b^{5}}$

2. Simplify. Assume that the variables represent positive real numbers.

   (a). $\sqrt{5x^8y^2}\sqrt{20x^6y^5}$

   (b). $\sqrt{21u^4v^5}\sqrt{3u^9v^2}$

3. Simplify. Assume that the variables represent positive real numbers.

   (a). $\sqrt[4]{6u^5}\cdot \sqrt[4]{27u^9}$

   (b). $\sqrt[3]{4v^6}\cdot \sqrt[3]{10v^2}$

   (c). $\sqrt[5]{8y^2}\cdot \sqrt[5]{16y^4}$

4. Multiply. Simplify the answer as much as possible.

   (a). $(8\sqrt{2}+\sqrt{7})({9\sqrt{7}-3\sqrt{2}})$

   (b). $(\sqrt{x}-2\sqrt{2})^2$

   (c). $(2\sqrt{y}-\sqrt{5})(2\sqrt{y}+\sqrt{5})$

5. Rationalize the denominator and simplify. Assume the variables represent positive real numbers.

   (a). $\dfrac{10}{\sqrt{5y}}$

   (b). $\sqrt{\dfrac{7x}{3}}$

6. Rationalize the denominator and simplify.

   (a). $\dfrac{-4}{3\sqrt{5}-4}$

   (b). $\dfrac{7}{5+2\sqrt{3}}$

Answer key

1. (a). $5y^2\sqrt{2}$

   (b).$6b^6\sqrt{b}$

2. (a). $10x^7y^3\sqrt{y}$

   (b). $3u^6v^3\sqrt{7uv}$

3. (a). $3u^3 \sqrt[4]{2u^2}$

   (b). $2v^2 \sqrt[3]{5v^2}$

   (c). $2y \sqrt[5]{4y}$

4. (a). $15+69\sqrt{14}$

   (b). $x-4\sqrt{2x}+8$

   (c). $4y-5$

5. (a). $\dfrac{10\sqrt{5y}}{5y}$

   (b). $\dfrac{\sqrt{21x}}{3}$

6. (a). $\dfrac{-12\sqrt{5}-16}{29}$

   (b). $\dfrac{35-14\sqrt{3}}{13}$