288. ${{2}^{3}}\cdot {{4}^{2}}\cdot {{\left( \sqrt{8} \right)}^{{2}/{3}\;}}$ es igual a

a) ${{2}^{8}}$

b) $\sqrt[3]{16}$

c) ${{2}^{6}}$

Teniendo en cuenta las propiedades de las potencias de números reales:

${{2}^{3}}\cdot {{4}^{2}}\cdot {{\left( \sqrt{8} \right)}^{{2}/{3}\;}}\ \ ={{2}^{3}}\cdot {{\left( {{2}^{2}} \right)}^{2}}\cdot {{\left( {{\left( 8 \right)}^{\frac{1}{2}}} \right)}^{\frac{2}{3}}}\ \ =\ \ {{2}^{3}}\cdot {{2}^{2\times 2}}\cdot {{8}^{\frac{1}{2}\times \frac{2}{3}}}\ \ =$

$=\ \ {{2}^{3}}\cdot {{2}^{4}}\cdot {{8}^{\frac{1}{3}}}\ \ =\ \ {{2}^{3+4}}\cdot {{\left( {{2}^{3}} \right)}^{\frac{1}{3}}}\ \ =\ \ {{2}^{7}}\cdot {{2}^{3\times \frac{1}{3}}}\ \ =$

$=\ \ {{2}^{7}}\cdot {{2}^{1}}\ \ =\ \ {{2}^{7+1}}\ \ =\ \ {{2}^{8}}$