14)
[tex]\displaystyle \\
\left(3^{ \frac{1}{2}} -2^{ \frac{1}{2} \right)\left(3^{ \frac{1}{2}} +2^{ \frac{1}{2} \right) = \left(3^{ \frac{1}{2}} \right)^2 - \left(2^{ \frac{1}{2}} \right)^2 =3^{ \frac{1}{2} \times 2} - 2^{ \frac{1}{2} \times 2} = 3 - 2 =\boxed{1} \\ \\ \\ \left(x^{ \frac{1}{2}} +y^{ \frac{1}{2} \right)\left(x^{ \frac{1}{2}} -y^{ \frac{1}{2} \right) = \left(x^{ \frac{1}{2}} \right)^2 - \left(y^{ \frac{1}{2}} \right)^2 =x^{ \frac{1}{2} \times 2} - y^{ \frac{1}{2} \times 2} =\boxed{x-y}[/tex]
15)
[tex]a) \\ \displaystyle \\
2^{ \frac{2}{x} }= \sqrt[3]{2} \\ \\
2^{ \frac{2}{x} }=2^{ \frac{1}{3} } \\ \\
\frac{2}{x} = \frac{1}{3} \\ \\
x = \frac{2 \cdot 3}{1} = \boxed{6} \\ \\ \\
c) \\
\sqrt{2}\cdot 2^{\frac{2}{x}}=\sqrt[3]{2}\\\\
2^{\frac{1}{2}}\cdot2^{\frac{2}{x}}}=\sqrt[3]{2}\\\\
2^{ \frac{1}{2} \cdot \frac{2}{x} } =2^{ \frac{1}{3}} \\ \\
\frac{1}{2} \cdot \frac{3}{x} = \frac{1}{3} \\ \\
\frac{3}{x} = \frac{2}{3} \\ \\
x = \frac{3 \cdot 3}{2} =\boxed{\frac{9}{2} }[/tex]
[tex]e) \\ \displaystyle \\
\Big( \sqrt{2} \cdot \sqrt[3]{2} \Big)^{ \frac{x+1}{3} }=\Big(\sqrt[3]{4} \Big)^{ \frac{1}{3} } \\ \\
\Big( 2^{ \frac{1}{2} } \cdot 2^{ \frac{1}{3} } \Big)^{ \frac{x+1}{3} }=\Big( 2^{2\cdot \frac{1}{3} } \Big)^{ \frac{1}{3} } \\ \\
\Big( \frac{1}{2} + \frac{1}{3} \Big) \cdot \frac{x+1}{3} = 2\cdot \frac{1}{3}\cdot \frac{1}{3} \\ \\
\frac{5}{6} \cdot \frac{x+1}{3} = 2\cdot \frac{1}{3}\cdot \frac{1}{3} \\ \\
\frac{5x+5}{18} = \frac{2}{9}
[/tex]
[tex]\displaystyle \\
\frac{5x+5}{18} = \frac{2}{9} \\ \\
5x+5 = \frac{2 \cdot 18}{9} \\ \\
5x+5 = 4 \\ \\
5x = 4 - 5 \\ \\
5x= -1 \\ \\
x = \frac{-1}{5} = \boxed{-\frac{1}{5} }[/tex]
