Question

COROANA+PUNCTE!⚠️⚠️⚠️ROG RASPUNSURI SERIOASE ​

Answer 1

Răspuns:

Explicație pas cu pas:

$N = [(2x)*(2x)^2*(2x)^3* ... *(2x)^{1000}] : [(-4x)*(-4x)^3*(-4x)^5 * ... * (-4x)^{1001}] =$

$= (2x)^{(1 + 2 + 3+ ... + 1000)} : (-4x)^{(1 + 3 + 5 + ... + 1001)} =$

$= (2x)^{\frac{1000*(1000+1)}{2} } : (-4x)^{\frac{501*(1001 + 1 )}{2} } =$

$= (2x)^{500500} : (-4x)^{251001} =$

$= 2^{500500}*x^{500500} : [(-4)^{251001}*x^{251001}] =$

$= 2^{500500}*x^{500500} : [-(4)^{251001}*x^{251001}] =$

$= 2^{500500}*x^{500500} : [-(2^2)^{251001}*x^{251001}] =$

$= 2^{500500}*x^{500500} : [-2^{2*251001}*x^{251001}] =$

$= 2^{500500}*x^{500500} : [-2^{502002}*x^{251001}] =$

$=-2^{(500500-502002)}*x^{(500500-251001)}=$

$=-2^{-1502}*x^{249499} <0$

oricare ar fi x ∈ N , deci nu este patrat perfect.

Answer 2

$\it N=[(2x)\cdot(2x)^2\cdot(2x)^3\cdot\ ...\ \cdot(2x)^{1000}]:[(-4x)\cdot(-4x)^3\cdot(-4x)^5\cdot\ ...\ \cdot(-4x)^{999}]=\\ \\ \\ =(2x)^{1+2+3+\ ...\ +1000}:(-4x)^{1+3+5+\ ...\ +999}=(2x)^{500\cdot1001}:(-4x)^{500\cdot500}=\\ \\ =(2^{500\cdot1001}:2^{500\cdot1000})\cdot(x^{500\cdot1001}:x^{500\cdot500})=2^{500}\cdot x^{500\cdot501}=\\ \\ \\ =(2^{250}\cdot x^{250\cdot501})^2$