[tex]\displaystyle\\
\texttt{c)}\\\\
\left\{\left[\left(\frac{2}{3}\right)^{1007}\times\left(\frac{2}{3}\right)^3\right]^2\times \left(\frac{2}{3}\right)^{2000}\right\}\times \left(\frac{2}{3}\right)^{4000}=\\\\\\
=\left\{\left[\left(\frac{2}{3}\right)^{1007+3}\right]^2\times \left(\frac{2}{3}\right)^{2000}\right\}\times \left(\frac{2}{3}\right)^{4000}=\\\\\\
=\left\{\left[\left(\frac{2}{3}\right)^{1010}\right]^2\times \left(\frac{2}{3}\right)^{2000}\right\}\times \left(\frac{2}{3}\right)^{4000}=
[/tex]
[tex]\displaystyle\\
=\left\{\left(\frac{2}{3}\right)^{1010\times 2}\times \left(\frac{2}{3}\right)^{2000}\right\}\times\left(\frac{2}{3}\right)^{4000}=\\\\
=\left\{\left(\frac{2}{3}\right)^{2020}\times \left(\frac{2}{3}\right)^{2000}\right\}\times\left(\frac{2}{3}\right)^{4000}=\\\\
=\left(\frac{2}{3}\right)^{2020+2000}\times\left(\frac{2}{3}\right)^{4000}=\\\\
=\left(\frac{2}{3}\right)^{4020}\times\left(\frac{2}{3}\right)^{4000}
=\left(\frac{2}{3}\right)^{4020+4000}=\boxed{\left(\frac{2}{3}\right)^{8020}}[/tex]
