Question

$( - 2) \times (2) ^{2} \times ( - 2) { }^{3} \times ( - 2) { }^{4} \times ..... \times ( - 2) {}^{100} \div {4}^{2525}$
Cine poate sa îmi facă problema aceasta folosind suma lui gauss? ​

Answer 1

Răspuns:

$\large \bf( - 2)^{1} \cdot ( - 2)^{2} \cdot( - 2)^{3} \cdot ( - 2)^{4} \cdot ....\cdot ( - 2)^{100} : {4}^{2525} =$

$\large \bf( - 2)^{1 + 2 + 3 + 4 + .... + 100} : {4}^{2525} =$

Aplicam suma lui Gauss pentru a afla exponentul lui (-2)

100 • (100 + 1) : 2 = 50 • 101 = 5050

$\large \bf( - 2)^{5050} : {4}^{2525} =$

$\large \bf( - 2)^{5050} : (2^{2})^{2525} =$

$\large \bf 2^{5050} : 2^{2 \cdot 2525} =$

$\large \bf 2^{5050} : 2^{5050} =$

$\large \bf 2^{5050 - 5050}={2}^{0} = \boxed{ \bf 1}$

#copaceibrainly

Answer 2

$\it (-2)\cdot(-2)^2\cdot(-2)^3\cdot\ ...\ \cdot(-2)^{100}:4^{2525}=(-2)^{1+2+3+\ ...\ +100}\ :\ (2^2)^{2525}=\\ \\ \\ (-2)^{5050}:2^{5050}=2^{5050}:2^{5050}=1\\ \\ \\\mathbf{1+2+3+\ ...\ +100=\dfrac{100\cdot101}{2}=50\cdot101=5050}$