सरल कीजिए :

$(i)$ $2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}$

$(ii)$ $\left(\frac{1}{3^{3}}\right)^{7}$

$(iii)$ $\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}$

$(iv)$ $7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}$

$(i)$  $2^{\frac{2}{3:}} \times 2^{\frac{1}{5}}=2^{\frac{2}{3}+\frac{1}{5}}=2^{13 / 15}$

$\left[\because \frac{2}{3}+\frac{1}{5}=\frac{10+3}{15}=\frac{13}{15}\right]$

$(ii)$   $\left(3^{\frac{1}{3}}\right)^{7}=\left(3^{-3}\right)^{7} $ $=3^{-3 \times 7}=3^{-21}$ $\left[\because a^{\frac{1}{n}}=a^{-n}\right]$

$(iii)$ $\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}=11^{\frac{1}{2}-\frac{1}{4}}=11^{\frac{1}{4}}\left[\because \frac{1}{2}-\frac{1}{4}=\frac{2-1}{4}=\frac{1}{4}\right]$

$(iv)$ $7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}=(7 \times 8)^{\frac{1}{2}}=(56)^{\frac{1}{2}}$