Answer
(a) $\frac{1}{3}$
(b) $9$
(c) $18x^{12}$
(d) $\frac{x^3}{8}$
(e) $3a^3$
(f) $\frac{\sqrt[6]{a}}{\sqrt[12]{b}}$
Work Step by Step
(a) $\frac{\sqrt[3]{4}}{\sqrt[3]{108}}=\sqrt[3]{\frac{4}{108}}=\sqrt[3]{\frac{1}{27}}=\frac{1}{3}$
(b) $27^{2/3}=(3^3)^{2/3}=3^{2/3\cdot 3}=3^2=9$
(c) $2x^2(3x^5)^2=2x^2\cdot 3^2(x^5)^2=2x^2\cdot 9x^{5\cdot 2}=2\cdot 9\cdot x^{2+2\cdot 5}=18x^{12}$
(d) $(2x^{-2})^{-3}x^{-3}=(2x^{-2}\cdot x)^{-3}=(2x^{-2+1})^{-3}=(2x^{-1})^{-3}=(\frac{2}{x})^{-3}=(\frac{x}{2})^3=\frac{x^3}{x^3}=\frac{x^3}{8}$
(e) $\frac{3a^{3/2}\cdot a^{1/2}}{a^{-1}}=3a^{3/2+1/2-(-1)}=3a^{2+1}=3a^3$
(f) $\frac{\sqrt{a\sqrt{b}}}{\sqrt[3]{ab}}=\frac{\sqrt{ab^{1/2}}}{(ab)^{1/3}}=\frac{(ab^{1/2})^{1/2}}{a^{1/3}b^{1/3}}=\frac{a^{1/2}b^{1/2\cdot 1/2}}{a^{1/3}b^{1/3}}=\frac{a^{1/2}b^{1/4}}{a^{1/3} b^{1/3}}=a^{1/2-1/3}b^{1/4-1/3}=a^{1/6}b^{-1/12}=\frac{\sqrt[6]{a}}{\sqrt[12]{b}}$
