Question

c) {[( 2/3)^1007x(2/3^3]^2x(2/3)^2000}x(2/3)^4000

Answer 1

   
[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]