Question

Calculati: a) 2 supra 3 • 2 supra 3 la puterea 2 • 2 supra 3 la puterea 3 • ...• 2 la puterea 100 : 0,(6) la puterea 5048. Multumesc!

Answer 1

$\displaystyle\\ \left(\frac{2}{3}\right)^1+\left(\frac{2}{3}\right)^2+\left(\frac{2}{3}\right)^3+...+ \left(\frac{2}{3}\right)^{100}:(0,6)^{5048}=\\ =\left(\frac{2}{3}\right)^{1+2+2+...+100}:\left(\frac{6}{9}\right)^{5048}=\\ =\left(\frac{2}{3}\right)^{\frac{\bf100\times101}{\bf 2}}:\left(\frac{2}{3}\right)^{5048}=\\ =\left(\frac{2}{3}\right)^{50\times101}:\left(\frac{2}{3}\right)^{5048}=\\ =\left(\frac{2}{3}\right)^{5050}:\left(\frac{2}{3}\right)^{5048}=\\ \left(\frac{2}{3}\right)^{5050-5048}=\frac{4}{9}$