[tex]\{[( \frac{2}{3})^{1007}\;\cdot \; ( \frac{2}{3})^{3}]^2-( \frac{2}{3})^{2000}\}\;\cdot \;( \frac{2}{3})^{4000}= \\ \\ =\{[( \frac{2}{3})^{1007+3}]^2-( \frac{2}{3})^{2000}\}\;\cdot \;( \frac{2}= \\ \\ =\{[( \frac{2}{3})^{1010}]^2-( \frac{2}{3})^{2000}\}\;\cdot \;( \frac{2}{3})^{4000}= \\ \\ =\{( \frac{2}{3})^{1010\;\cdot\;2}-( \frac{2}{3})^{2000}\}\;\cdot \;( \frac{2}{3})^{4000}= \\ \\ =\{( \frac{2}{3})^{2020}-( \frac{2}{3})^{2000}\}\;\cdot \;( \frac{2}{3})^{4000}= [/tex]
[tex]=( \frac{2}{3})^{2020}\;\cdot \;( \frac{2}{3})^{4000}-( \frac{2}{3})^{2000}\;\cdot \;( \frac{2}{3})^{4000}= \\ \\ =( \frac{2}{3})^{2020+4000}-( \frac{2}{3})^{2000+4000}= \\ \\=\boxed{( \frac{2}{3})^{6020}-( \frac{2}{3})^{6000}}= \boxed{( \frac{2}{3})^{6000}[( \frac{2}{3})^{20}-1]}[/tex]
