Exercises - Radicals

  1. Simplify:

    1. $\displaystyle{\sqrt[4]{81} + \sqrt[5]{-32}}$   $\ans{\displaystyle{1}}$

    2. $\displaystyle{(\sqrt{3} - \sqrt{7})(\sqrt{3} + \sqrt{7})}$   $\ans{\displaystyle{-4}}$

    3. $\displaystyle{(5x^2 - \sqrt{2})^2}$   $\ans{\displaystyle{25x^4 - 10x^2 \sqrt{2} + 2}}$

    4. $\displaystyle{8\sqrt{5} + \frac{25}{\sqrt{5}}}$   $\ans{\displaystyle{13\sqrt{5}}}$

    5. $\displaystyle{3\sqrt{32} - 2\sqrt{18} - \sqrt{8}}$   $\ans{\displaystyle{4\sqrt{2}}}$

    6. $\displaystyle{\frac{\sqrt[3]{(x+1)^4} \sqrt{(x+1)^3}} {\sqrt[6]{(x+1)^5}}}$   $\ans{\displaystyle{(x+1)^2}}$

    7. $\displaystyle{\frac{\sqrt[3]{(a+b)^5} \cdot \sqrt[4]{(a+b)^2}}{\sqrt{(a+b)^3}}}$   $\ans{\displaystyle{\sqrt[3]{(a+b)^2}}}$

    8. $\displaystyle{\sqrt[4]{\frac{x^8 y^4 z^{16}}{w^8}}}$   $\ans{\displaystyle{\frac{x^2 y \,z^4}{w^2}}}$

    9. $\displaystyle{\sqrt[3]{a^4} - \frac{\sqrt[3]{a^2}}{\sqrt[3]{a^{-1}}}+\sqrt[3]{\frac{27}{a^{-1}}}}$   $\ans{\displaystyle{\begin{array}{l}a \sqrt[3]{a} - a + 3\sqrt[3]{a}, \quad \textrm{or}\\\\(3+a)\sqrt[3]{a} - a\end{array}}}$

  2. Rationalize the denominator:

    1. $\displaystyle{\frac{\sqrt[3]{12}}{\sqrt[3]{9}}}$   $\ans{\displaystyle{\frac{\sqrt[3]{36}}{3}}}$

    2. $\displaystyle{\frac{5}{\sqrt[3]{2xy^2}}}$   $\ans{\displaystyle{\frac{5\sqrt[3]{4x^2y}}{2xy}}}$

    3. $\displaystyle{\frac{2\sqrt{z}}{\sqrt{z} + 7}}$   $\ans{\displaystyle{\frac{2z - 14\sqrt{z}}{z-49}}}$

    4. $\displaystyle{\frac{2-a}{\sqrt{b}-\sqrt{5}}}$   $\ans{\displaystyle{\frac{2\sqrt{b}+2\sqrt{5}-a\sqrt{b}-a\sqrt{5}}{b-5}}}$

    5. $\displaystyle{\frac{\sqrt[3]{xy}}{\sqrt[3]{9ab^2}}}$   $\ans{\displaystyle{\frac{\sqrt[3]{3a^2bxy}}{3ab}}}$

    6. $\displaystyle{\frac{2a}{3\sqrt{a} + 2\sqrt{b}}}$   $\ans{\displaystyle{\frac{2a(3\sqrt{a} - 2\sqrt{b})}{9a-4b}}}$

  3. Convert to exponential notation and simplify:

    1. $\displaystyle{\sqrt[6]{\frac{a^{12} b^{18}}{3^6}}}$   $\ans{\displaystyle{\frac{a^2 b^3}{3}}}$

  4. Write as a single radical and simplify:

    1. $\displaystyle{\frac{\sqrt{(2x-3)^3} \cdot \sqrt[4]{2x-3}}{\sqrt[3]{(2x-3)^4}}}$   $\ans{\displaystyle{\sqrt[12]{(2x-3)^5}}}$

  5. Simplify, writing in radical notation if appropriate:

    1. $\displaystyle{\left( \frac{d^{-2/3}}{9b^{-2}} \right)^{-1/2}}$   $\ans{\displaystyle{\frac{3 \sqrt[3]{d}}{b}}}$

    2. $\displaystyle{\frac{\sqrt{9x^2y^{-1}} \sqrt{2^{-1} x^{-1} y^2}}{\sqrt{2x^2 y^3}}}$   $\ans{\displaystyle{\frac{3\sqrt{x}}{2xy}}}$

    3. $\displaystyle{\frac{ \left( \sqrt{x} + \sqrt{x^3} \right)^2}{2^0 \, x \, \sqrt[17]{\sqrt[3]{x} \cdot \sqrt{x^5}}}}$   $\ans{\displaystyle{\frac{(1+x)^2}{\sqrt[6]{x}}}}$

    4. $\displaystyle{\frac{-1}{\sqrt{51} \, - \, \sqrt{6}} \, + \, \sqrt[3]{\frac{2}{1215}} \, - \, \frac{1}{\sqrt[3]{18} + 3} \, + \, \frac{\sqrt{3}}{9\sqrt{5}}}$   $\ans{\displaystyle{\frac{-\sqrt{51} - \sqrt{6} + \sqrt[3]{150} - 3 \sqrt[3]{12} + 3 \sqrt[3]{18} - 9 + \sqrt{15}}{45}}}$