Question

Chứng minh các hằng đẳng thức:

a, \(\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)=2^{32}-1\)

b,\(100^2+103^2+105^2+94^2=101^2+98^2+96^2+107^2\)

giúp mk

Answer 1

a) Đặt A = (2 + 1)(2² + 1)(2⁴ + 1 )(2⁸ +1)( 2¹⁶ +1 )

=> A = ( 2² - 1 ) (2² + 1)(2⁴ + 1 )(2⁸ +1)( 2¹⁶ +1 )

=> A = (2⁴ - 1)(2⁴ + 1 )(2⁸ +1)( 2¹⁶ +1 )

=> A = (2⁸ - 1)(2⁸ +1)( 2¹⁶ +1 )

=> A= (2¹⁶ -1 ) (2¹⁶ + 1) = 2³² - 1 => đpcm

Answer 2

b)

<=> \(\left(100^2-98^2\right)+\left(103^2-101^2\right)+\left(105^2-107^2\right)+\left(94^2-96^2\right)\) = 0

<=> \(\left(100-98\right)\left(100+98\right)+\left(103-101\right)\left(103+101\right)\)+ (105 -107)(105+107) + (94 - 96)(96 + 94) = 0

<=> \(2.198+2.204-2.212-2.190\) = 0

<=> \(2\left(198+204-212-190\right)=0\)

<=> \(\left(198-190\right)+\left(204-212\right)=0\)

<=> \(-8+8=0\) (luôn đúng) => đpcm

P/s: đây ko phải bài lớp 10 đâu!