Question

a) (3 supra 4) la puterea 100 x ( 2 supra 5 ) la puterea 200 x ( 25 supra 3 ) la puterea 300

Answer 1

 

$\displaystyle\bf\\\left(\frac{3}{4}\right)^{100}\times\left(\frac{2}{5}\right)^{200}\times\left(\frac{25}{3} \right)^{300}=\\\\\\=\frac{3^{100}}{4^{100}}\times\frac{2^{200}}{5^{200}}\times\frac{25^{300}}{3^{300}}=\\\\\\=\frac{3^{100}}{\Big(2^2\Big)^{100}}\times\frac{2^{200}}{5^{200}}\times\frac{\Big(5^2\Big)^{300}}{3^{300}}=\\\\\\=\frac{3^{100}}{2^{2\times100}}\times\frac{2^{200}}{5^{200}}\times\frac{5^{2\times300}}{3^{300}}=$

.

$\displaystyle\bf\\=\frac{3^{100}}{2^{200}}\times\frac{2^{200}}{5^{200}}\times\frac{5^{600}}{3^{300}}=\\\\\\=\frac{3^{100}\times2^{200}\times5^{600}}{2^{200}\times5^{200}\times3^{300}}=\\\\\\=3^{100-300}\times2^{200-200}\times5^{600-200}=\\\\=3^{-200}\times2^{0}\times5^{400}=\\\\=\boxed{\bf\frac{5^{400}}{3^{200}}}\\\\\\Explicati:\\\\a)~~2^0=1\\\\b)~~3^{-200}~a~trecut~la~numitor~si~semnul~"-"~a~disparut$